Appendix C

Answers

Acknowledgements

I am grateful to Milan Lukic <lmilan@shell.core.com> for pointing out several errors (now corrected): Equation (*) on page 3 was wrong, and the answer on page 7 to exercise 1.3.10 on page 29 contained a typo (Z_{t} instead of Z_{T} ). Also, the answers on page 9 to exercises 1.3.34–36 were garbled.

  Roger Sewell, <rfs@cambridgeconsultants.com>, found many errata,

both in the main text and in the answers. I tried to give credit at every instance, but I am not sure I succeeded; I know that on occasion I managed to introduce a new mistake in response to his suggestions. I want to state here that I am profoundly grateful for his faithful reports of errata he found.

2 Answers

Answers

1.1.1 See [9, page 41], or [5, page 293 ff.]: Consider a physical system, for instance an ear of corn, member of a (vast) corn field. An observable is a quantity about the ear that can be measured by a well–defined procedure. For example, the girth G, the number N of kernels, the length L, the weight W of our ear of corn are observables, measurable by applying a tape measure, by counting, or by weighing on a scale, respectively. The observables form an algebra and vector lattice E in the obvious way; for instance, G + 3W, LN, and G ∧ W are the observables having the procedures “measure girth and weight and add three times the latter to the former,” “multiply the length by the number of kernels,” and “take the smaller of girth and weight,” respectively. In fact, for a technical reason that will become transparent later let us consider complex observables (G + iW etc). They clearly form a commutative algebra A over the compex field ℂ. (Note the implicit requirement that different observables can be measured simultaneously – no quantum effects here.) For every observable Z = X + iY let Z^{∗} denote the observable whose value is the complex conjugate x − iy whenever a measurement of Z produces x+iy. Clearly (Z_{1}Z_{2})^{∗} = Z_{2}^{∗}Z_{1}^{∗}, Z^{∗∗} = Z, and (zZ)^{∗} = \overline{z}Z^{∗} for z ∈ ℂ and Z, Z_{1}, Z_{2} ∈ A. Let ∥Z∥ be the supremum of the possible values of |Z| = |X + iY | =^{def} \sqrt{X^{2} + Y^{2}} , taken over the ensemble to be represented (the corn field). Then restrict attention to the commutative normed ℂ-algebra A of those observables Z that have ∥Z∥ < ∞. The evident submultiplicativity ∥Z_{1}Z_{2}∥ ≤ ∥Z_{1}∥ · ∥Z_{2}∥ makes A a normed algebra. Its completion \overline{A} is easily seen to be a Banach algebra with a multiplicative unit I (the observable that produces the value 1 upon all measurements), where involution Z ↦ Z^{∗} and norm ∥ ∥ satisfy ∥Z_{1}Z_{2}∥ ≤ ∥Z_{1}∥ · ∥Z_{2}∥ and ∥Z^{∗}Z∥ = ∥Z∥^{2} . A Banach algebra with unit and involution as above is known as a unital C^{∗}-algebra.

Another common example of a commutative unital C^{∗}-algebra is the space C_{ℂ}(Ω) of continuous functions on a compact Hausdorff space Ω, with pointwise addition, multiplication, involution = complex conjugation, and equipped with the supremum norm. This example is typical. Namely, every commutative unital C^{∗}-algebra \overline{A} is of the form C_{ℂ}(Ω) for some compact Hausdorff space Ω. Let us take this fact for granted right now – its proof, which will also clarify the precise meaning of “is of the form,” can be found further down. Given this fact, the algebra and vector lattice closed under chopping E of real bounded observables is then identified with a dense subset Ê of the real part C_{ℝ}(Ω) of C_{ℂ}(Ω), for some compact Hausdorff Ω.

The probabilistic aspect in this story comes from the interest in the average of the various observables: what is the average weight of an ear of corn, and how can one estimate it without harvesting the whole field and toting it to the scales? A little reflection shows that the average is a linear positive functional on the observables E, with the average of the unit observable I being 1. Corresponding to it is a positive linear functional 𝔼 : Ê → ℝ with 𝔼[1] = 1. There is an immediate extension by continuity to all of C_{ℝ}(Ω). On C_{ℝ}(Ω), then, the average is (represented as) a positive Radon measure 𝔼 of total mass one. Such is automatically σ-additive. The extension theory of page 394 ff. applies and produces a σ-additive extension, called the expectation and again denoted by 𝔼. Its restriction to the integrable subsets F of Ω (see notation A.1.4) is the probability ℙ. The laws of large numbers allow the identification of ℙ[F] with a limiting frequency (and of 𝔼 as a limiting average). Most often (Ω, F, ℙ) is taken as the basic mathematical model for the probabilistic analysis, possibly because people might find the frequency of events intuitively more appealing than the average of measurements, and despite the difficulty of justifying the ad hoc requirement of σ-additivity of ℙ.

The latter is gone from the model (E, 𝔼).

The structure theorem for a unital commutative C^{∗}-algebra \overline{A} is left to be established. There will be several steps.

(i) If Z ∈ \overline{A} is invertible, then Z + H is invertible with inverse

(Z + H)^{−1} = Z^{−1} ∑_{k=0}^{∞}(−HZ^{−1})^{k} ,  (∗)

provided ∥H∥ < ∥Z^{−1}∥; simply multiply the evidently convergent sum on the right or left with Z + H, obtaining I in both cases. The invertible elements of \overline{A} therefore form an open set G. Since a proper ideal I is disjoint from G so is its closure \overline{I}, which therefore is proper as well. (ii) An element X ∈ \overline{A} is self–adjoint if X^{∗} = X. An ideal I is self–adjoint if it equals I^{∗} =^{def} {X^{∗} : X ∈ I}. By Zorn’s lemma, a proper self–adjoint ideal is contained in a maximal proper self–adjoint ideal M, which by (i) is closed. The quotient Ȧ =^{def} A/M is in the obvious way a C^{∗}-algebra and clearly contains no proper self–adjoint ideal. In fact, every non–zero element Ż ∈ Ȧ is invertible: if not, then the self–adjoint ideal Ȧ · Ż^{∗}Ż, which would not contain the unit İ, equals {˙0}, whence Ż^{∗}Ż = ˙0 and Ż = ˙0. In other words, Ȧ is a field. We shall see soon that “the C^{∗}-field” Ȧ equals ℂ. (iii) From the submultiplicativity of the norm, ∥Z^{n}∥^{1/n} ≤ ∥Z∥, so that ν(Z) =^{def} inf{∥Z^{n}∥^{1/n} : n ∈ ℕ} exists. It is not hard to see that ν(Z) = lim_{n→∞} ∥Z^{n}∥^{1/n}. Indeed, given an ϵ > 0, find an N ∈ ℕ with ∥Z^{N}∥^{1/N} < ν(Z) + ϵ. For n > N there are q, r with n = qN + r and 0 ≤ r < N. Then

∥Z^{n}∥^{1/n} ≤ ∥Z^{N}∥^{q/n}∥Z∥^{r/n} ≤ (ν(Z)+ϵ)^{qN/n}∥Z∥^{r/n} ⟶_{n→∞} ν(Z)+ϵ; hence

lim sup_{n→∞} ∥Z^{n}∥^{1/n} ≤ ν(Z) + ϵ ∀ ϵ > 0 and lim sup_{n→∞} ∥Z^{n}∥^{1/n} = ν(Z). Note that, for a self–adjoint element X ∈ \overline{A}, ∥X^{2}∥ = ∥X^{∗}X∥ = ∥X∥^{2} , hence

∥X∥ = ∥X^{2^{k}}∥^{2^{−k}} ⟶_{k→∞} ν(X), whence finally ν(X) = ∥X∥.

(iv) The resolvent of Z ∈ \overline{A} is the function ℂ ∋ z ↦ (zI − Z)^{−1} , defined on the open (by (i)) set υ_{Z} of z ∈ ℂ for which the inverse exists. The compact complement σ_{Z} of υ_{Z} is called the spectrum of Z. From the straightforward resolvent identity

(zI − Z)^{−1} − (z′I − Z)^{−1} = −(z − z′)(zI − Z)^{−1}(z′I − Z)^{−1}

it is clear that z ↦ (zI − Z)^{−1} is not only continuous, but even complex differentiable (analytic) on υ_{Z}. The observation that (zI − Z)^{−1} ⟶_{z→∞} 0 proves that the spectrum σ_{Z} is not empty: if it were, any continuous linear functional on \overline{A} applied to (zI −Z)^{−1} would produce an entire function that vanishes at infinity; such must vanish identically, and by the Hahn–Banach theorem we would arrive at the impossible consequence (zI − Z)^{−1} = 0 ∀ z. We apply this in the C^{∗}-field Ȧ = \overline{A}/M of (ii): Let 0 ≠ Ż ∈ Ȧ and z ∈ σŻ. Then zİ − Ż, not being invertible, must be zero: Ż = zI.

(v) By (∗), (zI − Z)^{−1} = (1/z) · ∑_{k=0}^{∞} z^{−k}Z^{k}. The time–honored root test, suitably adapted to series in \overline{A}, shows that the convergence radius of the series (in 1/z) on the right hand side is (ν(Z))^{−1} . That is to say, the circle [|1/z| = (ν(Z))^{−1}] = [|z| = ν(Z)] must contain a singularity of (zI − Z)^{−1} . From this we conclude that ν(Z) equals the spectral radius ρ(Z) of Z, which is defined as ρ(Z) = max{|z| : z ∈ σ_{Z}}. For a self–adjoint element X = X^{∗} ∈ \overline{A}, therefore,

∥X∥ = ν(X) = ρ(X) .  (∗∗)

(vi) Let Ω denote the collection of all linear multiplicative (ω(Z_{1}Z_{2}) = ω(Z_{1})ω(Z_{2})) functionals ω : \overline{A} → ℂ that take I to 1 and respect the involution (ω(Z^{∗}) = \overline{ω(Z)}), the ∗-characters. Such ω has norm 1. Indeed, let ∥Z∥ < 1. Then ∥Z^{∗}Z∥ < 1. Define a_{n} by \sqrt{1 − x} = ∑ a_{n}x^{n} for |x| < 1, and set Y =^{def} ∑ a_{n}(Z^{∗}Z)^{n}. Then I − Z^{∗}Z = Y^{∗}Y and so ω(I) − ω(Z^{∗}Z) = ω(Y^{∗}Y) > 0 and |ω(Z)| < 1. Given the topology of pointwise convergence, Ω is a compact Hausdorff space (see exercise A.2.13). For Z ∈ \overline{A} set

Ẑ(ω) =^{def} ω(Z) .

This defines a continuous function Ẑ : Ω → ℂ. The map Z ↦ Ẑ is clearly linear, multiplicative, turns involution on \overline{A} into complex conjugation on C_{ℂ}(Ω), and has Î = 1 and ∥Ẑ∥_{∞} ≤ ∥Z∥. It is left to be shown that it is in fact an isometry. To this end let Z ∈ \overline{A} and z = ∥Z∥. Then by (∗∗) we have |z|^{2} ∈ σ_{Z^{∗}Z}, and the proper self–adjoint ideal I =^{def} \overline{A} · (|z|^{2}I − Z^{∗}Z) is contained in a maximal proper self–adjoint ideal M, which is closed and gives rise to a bijective ∗-character ω̇ : Ȧ =^{def} \overline{A}/M → ℂ. Let ω be the composition of ω̇ with the quotient map \overline{A} → Ȧ. At this point ω ∈ Ω clearly |Ẑ(ω)|^{2} = ω(Z^{∗}Z) = |z|^{2} , so that ∥Ẑ∥_{∞} = |Ẑ(ω)| = ∥Z∥. Z ↦ Ẑ furnishes the desired linear isometric multiplicative involution– preserving identification of \overline{A} with C_{ℂ}(Ω). 1.2.4 (ii): For u ∈ ℕ let W^{u} be a countable uniformly dense subset of {w ∈ C[0, u] : w_{0} = 0} . Identify every path w in W^{u} with that path in 𝒞 which agrees with w on [0, u] and is constant thereafter. The collection ⋃_{u} W^{u} is plainly dense in 𝒞 in the topology of uniform convergence on compacta. 1.2.10 𝔼[W_{t}−W_{s}|F_{s}^{0}[W.]]=𝔼[W_{t}−W_{s}]=0. As 𝔼[−2W_{t}W_{s}|F_{s}^{0}[W.]]=−2W_{s}^{2} , 𝔼 [W_{t}^{2}−W_{s}^{2}|F_{s}^{0}[W.]] =𝔼 [(W_{t}−W_{s})^{2} |F_{s}^{0}[W.]] =𝔼 [(W_{t}−W_{s})^{2}] =t−s . 1.2.11 (i) ⟹ (ii): The independence of the increments gives

𝔼[M_{t}^{z} − M_{s}^{z}|F_{s}^{0}[X.]] = 𝔼[(e^{z(X_{t}−X_{s})−z^{2}(t−s)/2} − 1)e^{zX_{s}−z^{2}s/2}|F_{s}^{0}[X.]]

= 𝔼[(e^{z(X_{t}−X_{s})−z^{2}(t−s)/2} − 1)]𝔼[e^{zX_{s}−z^{2}s/2}] .

Now  𝔼[e^{zX_{s}−z^{2}s/2}] = \frac{1}{\sqrt{2π s}} ∫ e^{zx−z^{2}s/2}e^{−x^{2}/2s} dx

= \frac{1}{\sqrt{2π s}} ∫ e^{−(x−zs)^{2}/2s} dx = 1 ,

so 𝔼[M_{t}^{z} − M_{s}^{z}|F_{s}^{0}[X.]] = 0 .

(ii) ⟹ (iii) is obvious; and a computation as above shows that if (iii) is satisfied, then

𝔼[e^{iα(X_{t}−X_{s})+α^{2}(t−s)/2}|F_{s}^{0}[X.]] = 1 .

This clearly implies that the increment X_{t} −X_{s} is independent of all previous increments and that its characteristic function is e^{−α^{2}(t−s)/2} . This identifies X_{t} − X_{s} as a normal random variable with mean zero and variance t − s. 1.2.12 The Borel functions ϕ for which this equation holds form a vector space that contains the constants and is closed under pointwise limits of bounded sequences. Thanks to exercise A.3.5 it suffices to establish the claim for functions ϕ of the form ϕ(y) = exp(iαy). These form a multiplicative class that generates the Borel σ-algebra on ℝ. For such ϕ the right-hand side can be evaluated; by exercise A.3.45 on page 419 it equals

exp(−α^{2}(t − s)/2) · exp(iαW_{s}) .  (∗)

For any F_{s}^{0}[W.]-measurable bounded random variable F

𝔼 [e^{iαW_{t}}F ] = 𝔼 [e^{iα(W_{t}−W_{s})}F e^{iαW_{s}}] = 𝔼 [e^{iα(W_{t}−W_{s})}] 𝔼 [F e^{iαW_{s}}]

= e^{−α^{2}(t−s)/2} · 𝔼 [F e^{iαW_{s}}] .

Integrating (∗) against F yields the same. Thus the two F_{s}^{0}[W.]-measurable random variables of the statement are a.s. the same. 1.2.14 (ii): To study the joint law of the increments

t_{1} · W_{1/t_{1}} − t_{0} · W_{1/t_{0}}, …, t_{n} · W_{1/t_{n}} − t_{n−1} · W_{1/t_{n−1}}

use the characteristic function:

𝔼[e^{i ∑_{k=1}^{n} α_{k}·t_{k}W_{1/t_{k}} −t_{k−1}W_{1/t_{k−1}}} ]
= 𝔼[e^{iα_{1}·t_{1}W_{1/t_{1}} −t_{0}W_{1/t_{0}}} ] × 𝔼[e^{i ∑_{k=2}^{n} α_{k}·t_{k}W_{1/t_{k}} −t_{k−1}W_{1/t_{k−1}}} ]
= 𝔼[e^{iα_{1}·t_{0}W_{1/t_{1}} −t_{0}W_{1/t_{0}}} · e^{iα_{1}(t_{1}−t_{0})W_{1/t_{1}}} ] × 𝔼[…]
= 𝔼[e^{iα_{1}·t_{0}W_{1/t_{1}} −t_{0}W_{1/t_{0}}} ] · 𝔼[e^{iα_{1}(t_{1}−t_{0})W_{1/t_{1}}} ] × 𝔼[…]
= e^{−α_{1}^{2}·t_{0}^{2}·( \frac{1}{t_{0}} − \frac{1}{t_{1}} )} · 𝔼[e^{−α_{1}^{2}· \frac{(t_{1}−t_{0})^{2}}{t_{1}}} ] × 𝔼[…]
= e^{−α_{1}^{2}·(t_{1}−t_{0})} × 𝔼[…]

leads by induction to

𝔼[e^{i ∑_{k=1}^{n} α_{k}·t_{k}W_{1/t_{k}} −t_{k−1}W_{1/t_{k−1}}} ] = e^{− ∑_{k=1}^{n} α_{k}^{2}·(t_{k}−t_{k−1})} ,

which is the characteristic function of the joint distribution of the increments W_{t_{1}} − W_{t_{0}},…,W_{t_{n}} − W_{t_{n−1}} of a standard Wiener process. We conclude that W′_{t} =^{def} tW_{1/t} has independent stationary increments with law N(0, t − s). Import W′_{t} ⟶_{t→0} 0 from exercise 2.5.21 (ii). Setting W′_{0} =^{def} 0 we obtain a process W′ with almost surely continuous paths, a standard Wiener process (definition 1.2.3). 1.2.15 (i): Take the product of d independent copies of a standard onedimensional Wiener process. (ii): Every component W^{η} is a standard onedimensional Wiener process. (v): Exercise 1.2.10: 𝔼[W_{t}^{η}|F_{s}^{0}[W.]] = W_{s}^{η} and

𝔼 [W_{t}^{η}W_{t}^{θ} − W_{s}^{η}W_{s}^{θ}|F_{s}^{0}[W.]] = (t − s) · δ^{ηθ} .

Exercise 1.2.11: take z ∈ ℂ^{d} and replace zX by ⟨z|X_{t}⟩ = ∑_{η} z_{η}X_{t}^{η}. Exercise 1.2.12: f is now a Borel function on ℝ^{d}. The formula reads

𝔼 [ϕ(W_{t})|F_{s}^{0}[W.]]
= (2π(t − s))^{n/2} ∫_{−∞}^{+∞} ϕ(y) · e^{(^{−|y−W_{s}|^{2}/2(t−s)})} dy^{1} … dy^{d} ,

where | | is the euclidean norm on ℝ^{d}. The proof of exercise 1.2.12 given in this appendix applies literally, if the function ϕ(y) is read to mean ϕ(y) = exp(i⟨α|y⟩), etc. 1.2.16 In the proof of theorem 1.2.2 replace {ϕ_{n}} by a basis of the Hilbert space of Lebesgue square integrable functions on Ȟ. The estimate (1.2.2) that was used in the application on page 14 of Kolmogorov’s lemma A.2.37 can be replaced by

𝔼[|W_{z_{2}} − W_{z_{1}} |^{6} ] ≤ const · n^{3}∥z_{2} − z_{1} ∥_{∞}^{3} if ∥z_{i} ∥_{∞} ≤ n .

1.3.2 Apply theorem A.5.10. 1.3.6 Let A = [W_{∞}^{⋆} < c]. Then A lies in the intersection of the sets A_{n} = [|W_{n} − W_{n−1}| < 2c], each of which has the same probability q < 1. Since the A_{n} form an independent collection, ℙ[A] ≤ q^{N} for all N ∈ ℕ. Consequently

ℙ[W_{∞}^{⋆} < ∞] ≤ ∑_{c∈ℕ} ℙ[W_{∞}^{⋆} < c] = 0 .

1.3.10 Z_{T ∨t} − Z_{T} = (Z − Z^{T})_{t}. 1.3.15 (ii): [inf_{n≤N} T_{n}≤t] = ⋃_{n≤N}[T_{n}≤t]. [sup_{n∈ℕ} T_{n}>t]= ⋃_{n∈ℕ}[T_{n}>t] ∈ F_{t}. 1.3.16 (i): A ∈ F_{S} implies A ∩ [T ≤ t] = (A ∩ [S ≤ t]) ∩ [T ≤ t] ∈ F_{t}. (ii): [S < T] ∩ [T ≤ t] = ⋃_{q∈ℚ}[S ≤ q] ∩ [q < T ≤ t] ∈ F_{t} for all t and thus [S < T] ∈ F_{T} . Therefore [T ≤ S] = [S < T]^{c} ∈ F_{T} as well. [S < T] ∩ [S ≤ t] = (([S < T] ∩ [T ≤ t]) ∩ [S ≤ t]) ∪ ([T > t] ∩ [S ≤ t]) ∈ F_{t} for all t and thus [S < T] ∈ F_{S} and [T ≤ S] ∈ F_{S}. Finally, [S < T] = [S∧T < T] ∈ F_{S∧T} etc. 1.3.17 [[0, T))_{t} = [T > t] = [T ≤ t]^{c} belongs to F_{t} for all t precisely if T is a stopping time. In that case

[[0, T]]_{t} = [T ≥ t] = [T < t]^{c} = ( ⋃_{ℚ∋q<t}[T ≤ q])^{c} ∈ F_{t} .

All other stochastic intervals are differences of the two above. 1.3.18 [T_{A} ≤ t] = A ∩ [T ≤ t] ∈ F_{t} ∀ t. 1.3.20 ∀t ≥ 0 ∃k ∈ ℕ with k/n≤t<(k + 1)/n. Then [T^{(n)}≤ t] = [T ≤ k/n] ∈ F_{t}. 1.3.21 (i) and (ii): Adapt the proof of theorem 2.4.4 on page 69, but note that ∑_{n} ΔX_{T_{n}} · [[T_{n}]] will generally not converge. (iii): Define inductively

T^{j+1} = inf{s > T^{j} : |h(ΔX_{s})| ≥ ϵ} and set J^{ϵ} =^{def} ∑_{j} h(ΔX_{T^{j}})[[T^{j}, ∞)).

Clearly J.^{ϵ} is adapted, and so is J. = lim_{ϵ→0} J.^{ϵ}. 1.3.27 (i): Suppose the stopping times S, T agree almost surely. The set [S < T] = ⋃{[S < q < T] : q ∈ ℚ} belongs to A_{∞σ} and is negligible, so it is nearly empty. (ii): N = (N ∩ [T = ∞]) ∪ ⋃_{n} N ∩ [T ≤ n]. 1.3.28 Let X, Y be adapted right-continuous processes. The set where their paths differ: [X. ≠ Y.] = {ω ∈ Ω : ∃t with X_{t}(ω) ≠ Y_{t}(ω)} = ⋃_{q∈ℚ}[X_{q} ≠ Y_{q}] then belongs to A_{∞σ}; if X, Y are modifications of each other, it is negligible. 1.3.30 (i): If T is a stopping time, then [T < t] = ⋃_{n}[T ≤ (t − 1/n) ∨ 0] ∈ F_{t} . Conversely, if [T < t] ∈ F_{t} ∀ t > 0, then [T ≤ t] =

⋂_{n}[T < t + 1/n] ∈ ⋂_{n} F_{t+1/n}=F_{t+} = F_{t} .

(ii): [T < t] = ⋃ {[Z_{q} > λ] : ℚ ∋ q < t}. The T^{λ+} do not change if Z_{t} is replaced with Z_{t}^{+} =^{def} sup{Z_{s} : s ≤ t}. We may thus assume Z is increasing. If T^{λ+} < t, then Z_{t} > λ and consequently Z_{t} > μ for some μ > λ; that is to say, T^{μ+} ≤ t. Thus inf_{μ>λ} T^{μ+} = T^{λ+} , which says that λ ↦ T^{λ+} is right-continuous.

(iv): [T < t] = ⋃_{n}[T_{n} < t]; and A ∈ ⋂_{n} F_{T_{n}} ⟹ A ∩ [T_{n} < t] ∈ F_{t}∀n∀t ⟹ A ∩ [T < t] ∈ F_{t}∀t ⟹ A ∩ [T ≤ t]=A ∩ ⋂_{n}[T ≤ t + 1/n] ∈ F_{t+}=F_{t}.

(v): If X is left-continuous and adapted to F_{.+} , then X_{t−1/n} ∈ F_{(t−1/n)+}⊂F_{t} and consequently X_{t} = lim X_{t−1/n} ∈ F_{t}. Next let X be adapted to F. and progressively measurable for F_{.+} , and fix an instant t > 0. Evidently

X^{t} = X · [[0, t)) + X_{t} · [[t, ∞))

= lim_{1/t<n→∞} X · [[0, t − 1/n)) + X_{t} · [[t, ∞))

= lim_{n} X^{t−1/n} · [[0, t − 1/n)) + X_{t} · [[t, ∞)) .

The second summand is clearly measurable on B^{•}[0, ∞) ⊗ F_{t}. By the above X^{t−1/n} is measurable on B^{•}[0, ∞) ⊗ F_{(t−1/n)+} , which is contained in B^{•}[0, ∞) ⊗ F_{t} . The limit X^{t} is then also measurable on this product. Since this true for all t > 0, X is progressively measurable for F. . 1.3.31 Let ℙ ∈ 𝔓 and A, A^{(1)}, A^{(2)}, … ∈ F_{t}^{ℙ}. Let N^{ℙ} denote the ℙ-nearly empty sets. There are sets A_{ℙ}, A_{ℙ}^{(1)}, A_{ℙ}^{(2)}, … ∈ F_{t} with |A − A_{ℙ}| ∈ N^{ℙ} and |A^{(i)} − A_{ℙ}^{(i)}| ∈ N^{ℙ}, i = 1, 2, . . .. Evidently |A^{c} − A_{ℙ}^{c}| ∈ N^{ℙ}, showing that F_{t}^{ℙ} is closed under taking complements. Similarly, | ⋃_{i} A^{(i)} −⋃_{i} A_{ℙ}^{(i)}| ≤ ∑ |A^{(i)} − A_{ℙ}^{(i)}| ∈ N^{ℙ}, showing that F_{t}^{ℙ} is closed under taking countable unions: F_{t}^{ℙ} is a σ-algebra. Suppose A ∈ F_{∞}^{∗} is such that there exists an A_{ℙ} ∈ F_{t} with |A − A_{ℙ}| ∈ N^{ℙ}. Then A_{ℙ} \ A and A \ A_{ℙ} belong to F_{∞}^{∗} and are ℙ-nearly empty. Then A = (A_{ℙ} \ (A_{ℙ} \ A)) ∪ (A \ A_{ℙ}) belongs to the σ-algebra generated by F_{t} and the ℙ-nearly empty sets. This σ-algebra on the other hand is clearly contained in F_{t}^{ℙ}, since a ℙ-nearly empty set evidently belongs to it. 1.3.33 Consider the collection F̃_{t}^{ℙ} of random variables that satisfy this condition. If f^{(n)} ∈ F̃_{t}^{ℙ} converge pointwise on Ω to f and f_{ℙ}^{(n)} ∈ F_{t} differ only on a ℙ-nearly empty set from f^{(n)} , then set f_{ℙ} =^{def} lim sup f_{ℙ}^{(n)} . This random variable is measurable on F_{t}, and the set N =^{def} [f ≠ f_{ℙ}] is contained in the union of the sets [f^{(n)} ≠ f_{ℙ}^{(n)}], each of which is covered by a countable family of ℙ-negligible sets of A_{∞}. Then so is N: F̃_{t}^{ℙ} is sequentially closed, and therefore contains the σ-algebra generated by F_{t} and the ℙ-nearly empty sets, and every random variable measurable thereon. The converse F̃_{t}^{ℙ} ⊂ F_{t}^{ℙ} is obvious. 1.3.34 (ii): Suppose f satisfies this condition. Given ℙ ∈ 𝔓 we can find an F_{t}-measurable random variable f_{ℙ} such that N =^{def} [f ≠ f_{ℙ}] is ℙ-nearly empty. For any r ∈ ℝ, |[f < r] − [f_{ℙ} < r]| is a set of F_{∞}^{∗} contained in N, therefore [f < r] ∈ F_{t}^{ℙ}. Conversely, assume f is F_{t}^{𝔓}-measurable. For n ∈ ℕ and k ∈ ℤ set

f_{n} = ∑_{k=−∞}^{∞} k2^{−n} · [k2^{−n} < f ≤ (k + 1)2^{−n}]_{ℙ} .

These are F_{t}-measurable functions. Clearly lim sup f_{n} is F_{t}-measurable and differs only in a ℙ-nearly empty set from f. 1.3.35 (i): For every rational q > 0 let X′_{q} ∈ F_{q} be a random variable nearly equal to X_{q} (exercise 1.3.33), and set X′_{t} =^{def} lim inf{X′_{q} : ℚ ∋ q ↓ t}. X′ is clearly adapted to F_{.+} = F. . Outside the nearly empty set N =^{def} ⋃_{q}[X′_{q} ≠ X_{q}], X.^{′} is right-continuous and agrees with X. . The set [X′ ≠ X] is evidently evanescent. If X is a set, choose the X′_{q} idempotent. If X is increasing, define X′_{t} = lim_{ℚ∋q′↓t} sup_{q′≥q∈ℚ} X′_{q} instead.

(ii): The sufficiency of the condition is evident. For the necessity consider the F.^{ℙ}-adapted right-continuous decreasing process X =^{def} [[0, T)) (convention A.1.5 and exercise 1.3.17). Let X′ be a right-continuous F.-adapted set indistinguishable from X, and consider its “right edge”

T′ =^{def} inf{t ≥ 0 : X′_{t} = 0} = inf{t : (1 − X′)_{t} ≥ 1} .

This is an F.-stopping time (proposition 1.3.11), evidently nearly equal to T. If A ∈ F_{T}^{ℙ}, then the reduction T_{A} is a stopping time on F.^{ℙ}. Let T′ be an F.-stopping time nearly equal to T. Then A_{ℙ} =^{def} [T′ < ∞] meets the description of the second claim. 1.3.36 (i): Let A ∈ ⋂_{n} F_{t+1/n}^{𝔓}, and let ℙ ∈ 𝔓. There exist A^{(n)} ∈ F_{t+1/n} that is ℙ-nearly equal to A. Then A_{ℙ} =^{def} lim inf_{n} A^{(n)} ∈ F_{t+} is ℙ-nearly equal to A. Conversely, assume A ∈ F_{t+}^{𝔓}. Given ℙ ∈ 𝔓 we can find an A_{ℙ} ∈ F_{t+} that is ℙ-nearly equal to A. Therefore A ∈ F_{u}^{𝔓} for all u > t and A ∈ F_{t+}^{𝔓}. 1.3.42 Fix an instant t and a ℙ-negligible set A ∈ F_{t}. Then A ∩ [T > t] ∈ F_{T} is ℙ′-negligible for arbitrarily large stopping times T, and then so is A, provided we understand “arbitrarily large” to mean arbitrarily large with respect to ℙ′: for every ϵ > 0 and instant t there is a stopping time T with ℙ′[T ≤ t] < ϵ so that ℙ′ ≪ ℙ on F_{T} . 1.3.44 Let Ω be the half-line, let F_{∞} be the σ-algebra of Lebesgue measurable subsets of Ω, and let ℙ be the restriction of the normal law γ_{1} to F_{∞}. For F_{t} take the σ-algebra generated by the sets in F_{∞} that are contained in [0, t] and the interval (t, ∞). The pairs (F_{t}, ℙ) are all complete. If u > t, then {u} ∈ A_{∞} is ℙ-nearly empty, yet the outer measure that goes with the pair (F_{t}, ℙ_{|F_{t}}) does not annihilate {u}. 1.3.47 (i): F_{t}^{0} is the σ-algebra generated by the functions W_{s}, 0 ≤ s ≤ t (see page 15). Let t < u. We have to show that if f is a bounded function measurable on F_{t+} = ⋂_{t<t′<u} F_{t′}, then f is already measurable on F_{t}^{ℙ}. That is, we have to exhibit a function f^{ℙ} ∈ F_{t} that ℙ-nearly equals f. We let f^{ℙ} be the conditional expectation of f on F_{t}^{0} (theorem A.3.24 on page 407). It suffices to show that f^{ℙ} differs ℙ_{u}-negligibly from f (ℙ_{u} is of course the restriction of ℙ to F_{u}). In terms of the F_{t+}-measurable function g =^{def} f − f^{ℙ} this means that the measure μ =^{def} g · ℙ_{u} vanishes on F_{u}. Suppose we know that ∫ ϕ(ω) dμ(dω) = 0 for every function ϕ of the form (1.2.6):

ϕ(ω) = exp (i ∑_{k=1}^{L} r_{k}(W_{t_{k}}(ω) − W_{t_{k−1}}(ω))) ,  (∗)

r_{k} ∈ ℝ , 0 = t_{0} < t_{1} < … < t_{L} = u. Then we notice that the family of such functions ϕ forms a multiplicative class and apply exercise A.3.5: μ vanishes on all bounded functions measurable on the σ-algebra generated by the functions (∗); this σ-algebra is evidently F_{u}; and we are done. In proving that μ vanishes on the function ϕ of (∗), we may assume that t is among the t_{k}, say t = t_{K−1} , and by inserting another point strictly between t and the next instant and renaming that new instant t_{K} we may assume that t_{K} is as close to t as we please. We get

∫ ϕ dμ = 𝔼[g · ϕ dℙ]

= 𝔼[g · e^{(^{i ∑_{k≤K} r_{k}(W_{t_{k}} −W_{t_{k−1}} )})} · e^{(^{i ∑_{K<k≤L} r_{k}(W_{t_{k}} −W_{t_{k−1}} )})}]

= 𝔼[g · e^{(^{i ∑_{k≤K} r_{k}(W_{t_{k}} −W_{t_{k−1}} )})}] · ∏_{K<k≤L} 𝔼[e^{(ir_{k}(W_{t_{k}} −W_{t_{k−1}} ))}]

by A.3.45: = 𝔼[g · e^{(^{i ∑_{k≤K} r_{k}(W_{t_{k}} −W_{t_{k−1}} )})}] · ∏_{K<k≤L} e^{(^{−r_{k}^{2}(t_{k}−t_{k−1})/2})} .

We have used the fact that g is measurable on F_{t_{K}}^{ℙ}, and that the following increments are independent of this σ-algebra and of each other. Now as t_{K} ↓ t, the second factor in the last line stays bounded, and the first factor converges to

𝔼[g · exp (i ∑_{t_{k}≤t} r_{k}(W_{t_{k}} − W_{t_{k−1}}))] ,

which is zero, since the exponential is F_{t}-measurable.

(ii): The set [T^{±} = 0] belongs to F_{.+}^{0}[W] ⊆ F^{ℙ}[W]. The latter σ-algebra contains only sets nearly equal to either ∅ or Ω. Thus ℙ[T^{±} = 0] equals either zero or one. If ℙ[T^{+} = 0] = 0, then ℙ[T^{−} = 0] = 0 and vice versa – simply consider that T^{−} is “T^{+} for −W.” In that case T^{±} > 0 almost surely and W_{T^{±}∧t}^{2} = 0 for all t. From exercise 1.2.10 in conjunction with the optional stopping theorem 2.5.22 we see that then E[T^{±} ∧ t] = 0. In other words, we must have ℙ[T^{+} = 0] = ℙ[T^{−} = 0] = 1. 2.1.10 (i): If S is an elementary stopping time, then

∫ X dZ^{S} = ∫ X · [[0, S]] dZ  (∗)

for all X ∈ E. Let t > 0 and let T_{n} denote the stopping times of exercise 1.3.20. With S = T_{n} ∧ t (∗) gives

∫ X dZ^{T_{n}∧t} = ∫ X · [[0, T_{n} ∧ t]] dZ ,  X ∈ E .

As X ranges over the unit ball of E, the integrals on the right stay in some bounded subset of L^{p}, call it B. As n → ∞, the left-hand side converges to ∫ X dZ^{T ∧t}, due to the right-continuity of the paths. As X ∈ E_{1} varies, all these integrals stay in the closure of B, which is again a bounded set. That is to say, Z^{T ∧t} = (Z^{T} )^{t} is global L^{p}-integrator for all t, which means that Z^{T} is an L^{p}-integrator.

(ii): ∫ X dZ^{S∨T} = [S < T] · ∫ X dZ^{S} + [T ≤ S] · ∫ X dZ^{T} for X ∈ E. As X ranges over the unit ball of E, these integrals stay in a bounded set of L^{p}. This argument covers the “global” case. Replace in it S with S ∧ t and T with T ∧ t to get the “plain” case. 2.1.13 If such an extension is to exist, then Z must satisfy (RC-0). Namely, Z_{u} − Z_{t} equals ∫ ((t, u]] dZ, and if u ↓ t and consequently ((t, u]] ↓ ∅, this must converge to zero in L^{p}(ℙ), and thus a fortiori in measure. To see the necessity of (B-p), recall first what it means that a subset I of L^{p} is bounded: every neighborhood V of zero in L^{p} absorbs V; i.e., there is a scalar r such that I ⊂ r · V. Now if (B-p) were violated, there would exist a Y ∈ E_{+} such that the collection I of integrals ∫ X dZ of elementary integrands X in the order interval

[−Y, Y] =^{def} {X ∈ E : |X| ≤ Y }

were unbounded. For r = n there would be X_{n} ∈ [−Y, Y] with ∫ X_{n} dZ ∉ nV. The sequence (X_{n}/n) of elementary integrands would converge pointwise and dominatedly (by Y ) to zero, yet their integrals would stay outside V. 2.2.14 Use exercise 1.3.20 and the argument of proposition 2.2.11. 2.3.3 We use again the set S of 2.3.2 and set Z^{S} =^{def} sup{|Z |_{s} : s ∈ S}. For λ < ∥Z^{S} ∥_{[α]} , T =^{def} inf{s ∈ S : |Z |_{s} > λ} ∧ u is strictly less than u on [Z^{S} > λ], a set of probability > α where |Z |_{T} ≥ λ. We get the inequality

λ = λ∥[Z^{S} > λ]∥_{[α]} ≤ ∥|Z |_{T} ∥_{[α]} .

With the independent Bernoulli random variables ϵ_{1}, …, ϵ_{d} of theorem A.8.26,

|Z|_{T} ≤ K_{0}∥∑_{η} Z_{T}^{η} ϵ_{η}∥_{[κ_{0};τ]} ,
and with exercise A.8.16: λ ≤ K_{0}∥∥∑_{η} Z_{T}^{η} ϵ_{η}∥_{[γ;ℙ]}∥_{[ακ_{0}−γ;τ]} ,  γ < ακ_{0} .
Now  ∑_{η} Z_{T}^{η} ϵ_{η}(t) = ∫ ∑_{η} [[0, T]] dZ^{η} ,
and consequently  λ ≤ K_{0}∥ Z^{u} _{[γ]}∥_{[ακ_{0}−γ;τ]} = K_{0} Z^{u} _{[γ]} .

Now take the supremum over γ < ακ_{0} , λ < ∥Z^{S} ∥_{[α]} and S ⊂ [0, u] etc.

2.3.7 Let X = f_{0} ·[[0]]+∑_{n=1}^{N} f_{n} ·((t_{n}, t_{n+1}]] be an elementary integrand that vanishes past t and has |X| ≤ 1, as in equation (2.1.1) on page 46. Then

∫ X d|Z| = f_{0}|Z|_{0} + ∑_{n=1}^{N} f_{n}(|Z|_{t_{n+1}} − |Z|_{t_{n}})

= f_{0} sgn(Z_{0}) · Z_{0} + ∑_{n=1}^{N} f_{n} sgn(Z_{t_{n}})(Z_{t_{n+1}} − Z_{t_{n}})

+ ∑_{n=1}^{N} f_{n}(|Z_{t_{n+1}} | − |Z_{t_{n}} | − sgn(Z_{t_{n}})(Z_{t_{n+1}} − Z_{t_{n}}))

≤ ∫ XY dZ + A ,  (∗)

where  Y =^{def} sgn(Z_{0}) · [[0]] + ∑_{n=1}^{N} sgn(Z_{t_{n}}) · ((t_{n}, t_{n+1}]]

is an elementary integrand in E_{1} and

A =^{def} ∑_{n=1}^{N} (|Z_{t_{n+1}} | − |Z_{t_{n}} | − sgn(Z_{t_{n}})(Z_{t_{n+1}} − Z_{t_{n}}))

is a sum of positive random variables; indeed, since the absolute value function |.| is convex, |z_{2}| − |z_{1}| − sgn(z_{1})(z_{2} − z_{1}) ≥ 0 for any two reals z_{1}, z_{2} . For the choice X = [[0, t]] we actually get the equality |Z|_{t} = ∫ Y dZ +A, whence A = |Z|_{t} − ∫ Y dZ. Now (∗) stays when X is replaced by −X. Therefore

|∫ X d|Z|| ≤ |∫ XY dZ | + A ≤ |∫ XY dZ | + |∫ [[0, t]] dZ | − ∫ Y dZ ,

sum of three random variables that all have \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{p}-mean less than Z^{t} _{I^{p}}. 2.3.8 (ii) For an elementary \overline{F}.-stopping time \overline{T}, [[0, \overline{T}]]. ∘ \underline{R} ∈ E ^{1}. Conversely, if T is an elementary F.-stopping time, then for every one of its values t_{i} there is an \overline{A}_{i} ∈ \overline{F}_{t_{i}} so that [T = t_{i}] = \overline{A}_{i} ∘ \underline{R}. Then \overline{T} =^{def} ∑_{i} t_{i} · \overline{A}_{i} is an elementary stopping time on \overline{F}. with T = \overline{T} ∘ \underline{R} and [[0, T]]. = [[0, \overline{T}]]. ∘ \underline{R}. Taking differences and linear combinations as in equation (2.1.4) we see that E consists exactly of the processes \overline{X} ∘ \underline{R} with \overline{X} ∈ \overline{E}. The processes \overline{X} with \overline{X} ∘ \underline{R} ∈ P are evidently sequentially closed; thus \overline{P} ∘ \underline{R} ⊆ P – here \overline{P} denotes the predictables for \overline{F}. , of course. Conversely, the processes X of the form X = \overline{X} ∘ \underline{R} are also sequentially closed and contain E, so they exhaust P. (iv) If N ∈ \overline{F}_{t} is \overline{ℙ}-negligible, then N ∘\underline{R} = \underline{R}^{−1}(N) is ℙ-negligible; therefore the inverse image of a \overline{ℙ}-nearly empty set is ℙ-nearly empty, and \overline{X} ∘ \underline{R} is

1 See convention A.1.5 on page 364

(F., ℙ)-previsible provided \overline{X} is (\overline{F}., \overline{ℙ})-previsible.

(v) Equation (2.3.9) extends to \begin{aligned}⌈ \\ ⌈ & \overline{F} & ⌉ \\ & & ⌉ \end{aligned}_{\overline{Z}−p}^{∗} = \begin{aligned}⌈ \\ ⌈ & \overline{F} ∘ \underline{R} & ⌉ \\ & & ⌉ \end{aligned}_{Z−p}^{∗} in the two steps of Daniell’s up-and-down procedure. This leads directly to the remaining claims. 2.4.3 Hint: Suppose V _{t+} = inf{ V _{u} : u > t}> V _{t}+ϵ. Set U =^{def} {u > t : |V_{u} − V_{t}| < ϵ/2} and pick u_{1} ∈ U. There are {t = t_{0} < t_{1} < … < t_{I+1} = u_{1}} with ∑_{i} |V_{t_{i+1}} − V_{t_{i}}| > ϵ. Repeat this with u_{2} = t_{1} etc. and conclude that V_{t+} = ∞. 2.4.9 (i): If T^{λ} < τ, then I_{t} ≥ λ for some t < τ and consequently I_{τ} > λ and T^{λ+} ≤ τ. Thus T^{λ+} ≤ T^{λ}. The reverse inequality is obvious. (ii): [T^{Λ+} < t] = ⋃[T^{Λ^{(n)}+} < t], so it suffices to consider the case that Λ takes only countably many values λ_{n}. As [T^{λ_{n}+} < t] ∩ [Λ = λ_{n}] ∈ F_{t}, [T^{Λ+} < t] = ⋃[T^{Λ+} < t] ∩ [Λ = λ_{n}] ∈ F_{t}. 2.5.2 Since by exercise A.3.27 (v)

𝔼[M_{t}^{g}|F_{s}] = 𝔼[𝔼[g|F_{t}] |F_{s}] = 𝔼[g|F_{s}] = M_{s}^{g} ,  s < t ,

M^{g} is a martingale. Next, there exists a K ∈ ℝ such that the function g_{K} = (−K ∨ g ∧ K) has ∥g − g_{K} ∥_{L^{1}} < ϵ. Let G be any sub-σ-algebra of F_{∞}. Then 𝔼[g_{K}|G] lies between −K and K and has L^{1}-distance less than ϵ from g, since by Jensen’s inequality A.3.24

|𝔼[g|G] − 𝔼[g_{K}|G]| = |𝔼[g − g_{K}|G]| ≤ 𝔼[|g − g_{K}| |G] .

Thus the collection {𝔼[g_{K}|G] : G ⊂ F_{∞} } is uniformly integrable. 2.5.4 See exercises 1.2.10, 1.2.11, and 1.3.47. 2.5.5 Set g_{∞} =^{def} 𝔼[g| ⋁ 𝔉]. Let ϵ > 0 and consider the algebra ⋃ 𝔉. Its step functions are dense in L^{1}(⋁ 𝔉, ℙ). There is such a step function g^{ϵ} with ∥g_{∞} − g^{ϵ} ∥_{1} ≤ ϵ. It is measurable on some G ∈ 𝔉. For G ⊂ G′ ∈ 𝔉

∥g_{∞} − g^{G′} ∥_{1} ≤ ∥g_{∞} − g^{ϵ} ∥_{1} + ∥g^{ϵ} − g^{G′} ∥_{1} ≤ 2ϵ .

2.5.6 Without loss of generality we may assume that both f and f′ are bounded (how?). For every n ∈ ℕ let F_{n} be the σ-algebra generated by the sets [k2^{−n} < f ≤ (k + 1)2^{−n}], k = 0, 1, 2, …, and the ℙ-negligible subsets of F. Both f and f′ are measurable on the σ-algebra F_{∞} generated the F_{n}. M_{n} =^{def} 𝔼[f|F_{n}] and M′_{n} =^{def} 𝔼[f′|F_{n}] are uniformly integrable martingales on the filtration {F_{n}}_{n∈ℕ} (example 2.5.2). The hypothesis translates to M_{n} = M′_{n} ℙ-almost surely. Since M_{n} → f and M′_{n} → f′ in L^{1}(ℙ)-mean (exercise 2.5.5), f = f′ ℙ-almost surely. 2.5.12 (i): By corollary 2.5.11 this condition is necessary. To see that it is sufficient, let s < t be two instants, and let A ∈ F_{s}. Form the elementary stopping time s_{A} ∧ t which equals s on A and t off A (exercise 1.3.18). The given information 𝔼[M_{t} − M_{s_{A}}] = 0 can be rewritten as

∫_{A} 𝔼 [M_{t}|F_{s}] dℙ = ∫_{A} M_{s} dℙ .

(ii): Let M, N be supermartingales, 0 ≤ t and A ∈ F_{s}. Then

∫ M_{t} ∧ N_{t}·A dℙ = ∫ M_{t} ∧ N_{t}·[M_{s} < N_{s}]A dℙ + ∫ M_{t} ∧ N_{t}·[N_{s} ≤ M_{s}]A dℙ
≤ ∫ M_{t}·[M_{s} < N_{s}]A dℙ + ∫ N_{t}·[N_{s} ≤ M_{s}]A dℙ
≤ ∫ M_{s}·[M_{s} < N_{s}]A dℙ + ∫ N_{s}·[N_{s} ≤ M_{s}]A dℙ

= ∫ M_{s} ∧ N_{s}·[M_{s} < N_{s}]A dℙ + ∫ M_{s} ∧ N_{s}·[N_{s} ≤ M_{s}]A dℙ

= ∫ M_{s} ∧ N_{s}·A dℙ .

2.5.14 For the first statement apply exercise 1.3.35 on page 38. Next, since M is uniformly integrable it is L^{1}-bounded and M_{∞} =^{def} lim_{n} M_{n} exists pointwise. Again since M is uniformly integrable this limit is taken in L^{1}-mean (theorem A.8.6). 2.5.16 Set F_{t} = F_{n} and M_{t} = M_{n} for n ≤ t < n+1 and apply proposition 2.5.13. 2.5.21 (i): Use exercise 1.2.11 and lemma 2.5.18. (ii): From (i) ℙ[sup_{s<n} W_{s}/n>β/n+α/2]≤e^{−αβ}. With β=n^{2/3} and α=n^{−1/3}, the first Borel–Cantelli lemma gives ℙ[sup_{s<n} W_{s}/n > 2n^{−1/3} i.o.] = 0. This implies lim sup W_{t}/t ≤ 0, and lim inf W_{t}/t = − lim sup −W_{t}/t ≥ 0. 2.5.23 (ii): Let 0 ≤ s < t and A ∈ F_{s}. Then

𝔼[M_{t}^{T} · A] = 𝔼[M_{t}^{T} · A · [T ≤ s]] + 𝔼[M_{t}^{T} · A · [T > s]]
= 𝔼[M_{T ∧t} · A · [T ≤ s]] + 𝔼[M_{T ∧t} · A · [T > s]]

= 𝔼[M_{T ∧s} · A · [T ≤ s]] + 𝔼[M_{s∨T ∧t} · A · [T ∧t > s]]

by theorem 2.5.22:  = 𝔼[M_{T ∧s} · A · [T ≤ s]] + 𝔼[M_{s} · A · [T ∧t > s]]
= 𝔼[M_{T ∧s} · A · [T ≤ s]] + 𝔼[M_{T ∧s} · A · [T > s]]
= 𝔼[M_{T ∧s} · A] = 𝔼[M_{s}^{T} · A]
and  𝔼[M_{t}^{T} |F_{s}] = M_{s}^{T} .

(iii): Let M be a local martingale. Given a t ∈ ℝ_{+} and ϵ > 0 one can find a stopping time T with ℙ[T < t] < ϵ such that M^{T} is a martingale. Then T ∧t has ℙ[T ∧ t < t] < ϵ and reduces M to a uniformly integrable martingale. (iv): Let 0 ≤ s < t < ∞ and A ∈ F_{s}. There exist stopping times T_{n} that increase without bound and reduce M to martingales. For m ∈ ℕ set A_{m} = A ∩ [s < T_{m}]. Then A_{m} ∈ F_{s∧T_{m}} and by Fatou’s lemma A.8.7

𝔼[M_{t} A_{m}] ≤ lim inf_{n→∞} 𝔼[M_{t∧T_{n}} A_{m}]

by part (i):  = 𝔼[M_{s∧T_{m}} A_{m}]
since s < T_{m} on A_{m}:  = 𝔼[M_{s} A_{m}] ≤ 𝔼[M_{s} A] .
Hence  𝔼[M_{t} A] ≤ 𝔼[M_{s} A] .

For the last statement of (iv) write

𝔼[M_{S∨T}] = 𝔼[M_{S∨T} [S ≤ T]] + 𝔼[M_{S∨T} [S > T]]
= 𝔼[M_{T} [S ≤ T]] + 𝔼[M_{S}[S > T]]
by 1.3.16 (iv):  = 𝔼[𝔼[M_{T} |F_{S∧T}][S ≤ T]] + 𝔼[𝔼[M_{S}|F_{S∧T}][S > T]]
by part (i):  = 𝔼[M_{S∧T} [S ≤ T]] + 𝔼[M_{S∧T} [S > T]] = 𝔼[M_{S∧T}] = 𝔼[M_{0}]

2.5.25 For X = f_{0} · [[0]] + ∑_{n=1}^{N} f_{n} · ((t_{n}, t_{n+1}]] an elementary integrand as in the proof of theorem 2.5.24 we take the square root in

𝔼[(∫ X dW )^{2}] = 𝔼[(∑_{n=1}^{N} f_{n} · (W_{t_{n+1}} − W_{t_{n}}))^{2}]

= 𝔼[∑_{n=1}^{N} f_{n}^{2} · (W_{t_{n+1}} − W_{t_{n}})^{2}]

= 𝔼[∑_{n=1}^{N} f_{n}^{2} · (W_{t_{n+1}}^{2} − 2W_{t_{n+1}}W_{t_{n}} + W_{t_{n}}^{2})]

by 2.5.4 and 2.5.3:  = 𝔼[∑_{n=1}^{N} f_{n}^{2} · (W_{t_{n+1}}^{2} − W_{t_{n}}^{2})]

by 2.5.4:  = 𝔼[∑_{n=1}^{N} f_{n}^{2} · (t_{n+1} − t_{n})] = ∫ ∫ X_{s}^{2} ds dℙ .

For the second equality chose X = [[0, t]]. 2.5.31 From proposition A.8.24

A_{p}^{(2.5.6)} ≤ \{ \begin{matrix}(\frac{4p}{(p − 1)(2 − p)})^{1/p} & for 1 < p < 2, \\ 1 & for p = 2 \\ (\frac{3 · 8^{3/2} · p}{p − 3/2} + \frac{8^{3} · 3 · p}{3 − p})^{1/p} & for 2/3 < p < 3 \\ (\frac{4p}{p − 2})^{1− 1/p} & for 2 < p < ∞. \end{matrix}

(Since lim_{2≠p→2} A_{p} ≈ 57.8, this is an unsatisfactory formula; the problem arises to fashion a better one. Using complex interpolation one can show that A_{p} ≤ ( \frac{16}{p−1} )^{2(2−p)} for 1 < p < 2, which has at least the right limit behavior at p = 2) 2.5.32 There is a nearly empty set N_{1} off which the path ℚ ∋ q ↦ S_{q} has no oscillatory discontinuities (lemma 2.5.27 and lemma 2.3.1). Use this to define S′_{t} =^{def} lim_{q↓t} S_{q}, a supermartingale with right-continuous paths and adapted to F_{.+}^{𝔓}. For δ > 0 set T_{δ} =^{def} inf{t : S_{t} < δ}. This is a F_{.+}^{𝔓}-stopping time at which S′_{T_{δ}} ≤ δ. Let T_{δ}^{(n)} be the stopping times of exercise 1.3.20 and fix an instant t. By exercise 2.5.12, 𝔼[(S_{t} − S_{t∧T_{δ}^{(n)}}) · [T_{δ}^{(n)} < t]] ≤ 0, which in the limit produces 𝔼[S_{t} · [T_{δ} < t]] = 𝔼[S′_{t} · [T_{δ} < t]] ≤ 𝔼[S_{t∧T_{δ}} · [T_{δ} < t]] ≤ δ · ℙ[T_{δ} < t] ≤ δ and exhibits N_{2} =^{def} ⋂_{δ}[T_{δ} < t] as ℙ-nearly empty, inasmuch as S_{t} is almost surely strictly positive. Now if the restriction to ℚ of S.(ω) is not bounded away from zero on [0, t], then neither is S.^{′}(ω) and ω must lie in N_{2} : N =^{def} N_{1} ∪ N_{2} meets the description. 2.5.33 Let s < t and A ∈ F_{s}. There are stopping times U_{n} that converge almost surely to ∞ and reduce M to martingales, so that

𝔼[A · (M_{t}^{U_{n}} − M_{s}^{U_{n}})] = 0. Now M_{t}^{U_{n}} ⟶_{n→∞} M_{t} with |M_{t}^{U_{n}}| ≤ M_{t}^{⋆} ∈ L^{1}

when M is an L^{1}-integrator (see theorem 2.3.6 on page 63). Then 𝔼[A · (M_{t} − M_{s})] = 0, and we see M is a martingale. The second claim is established similarly. 3.1.2 Let S^{(n)}, T^{(n)} be the stopping times of exercise 1.3.20. Then clearly X^{(n)} =^{def} n · ((S^{(n)} ∧ n, T^{(n)} ∧ n]] are elementary integrands whose supremum H = ∞ · ((S, T]] ∈ E_{+}^{↑} majorizes |F |. It suffices to show that \begin{aligned}⌈ \\ ⌈ & H & ⌉ \\ & & ⌉ \end{aligned}_{Z−0}^{∗} ≤ ϵ. But this is evident: the stochastic integral f =^{def} ∫ X dZ of any X ∈ E with |X| ≤ H vanishes off [S < T] and therefore has \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{0} ≤ ϵ; the supremum of

such \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{0} is \begin{aligned}⌈ \\ ⌈ & H & ⌉ \\ & & ⌉ \end{aligned}_{Z−0}^{∗}.

3.1.3 This is immediate from exercise 2.5.25 and Daniell’s construction. 3.2.2 For every n there is a countable collection {X^{(n,k)}} in E whose pointwise supremum is H^{(n)} . The functions X^{(n)} =^{def} sup_{ν,k≤n} X^{(ν,k)} ≤ H^{(n)} belong to E and increase pointwise to sup_{n} H^{(n)} . Hence

\begin{aligned}⌈ \\ ⌈ & sup_{n} H^{(n)} & ⌉ \\ & & ⌉ \end{aligned}^{∗} = sup_{n} \begin{aligned}⌈ \\ ⌈ & X^{(n)} & ⌉ \\ & & ⌉ \end{aligned}^{∗} ≤ sup_{n} \begin{aligned}⌈ \\ ⌈ & H^{(n)} & ⌉ \\ & & ⌉ \end{aligned}^{∗} ≤ \begin{aligned}⌈ \\ ⌈ & sup_{n} H^{(n)} & ⌉ \\ & & ⌉ \end{aligned}^{∗} .

3.2.5 Suppose F is evanescent. Then the projection N of [F ≠ 0] on Ω is nearly empty. By the regularity of the filtration, X =^{def} N × [0, ∞) is an elementary integrand, and evidently \begin{aligned}⌈ \\ ⌈ & X & ⌉ \\ & & ⌉ \end{aligned}_{Z−p} = 0. The countable subadditivity of the mean implies that \begin{aligned}⌈ \\ ⌈ & ∞ · X & ⌉ \\ & & ⌉ \end{aligned}_{Z−p}^{∗} = 0, the solidity then

that \begin{aligned}⌈ \\ ⌈ & F & ⌉ \\ & & ⌉ \end{aligned}_{Z−p}^{∗} = 0.

3.2.11 The first statement was done in 3.2.7 if F is everywhere defined. 3.2.12 (ii): Given an ϵ > 0 find N ∈ ℕ so that ∑_{n≥N} \begin{aligned}⌈ \\ ⌈ & F_{n} & ⌉ \\ & & ⌉ \end{aligned}^{∗} < ϵ/2. Then find 0 < r < 1 so that ∑_{n<N} \begin{aligned}⌈ \\ ⌈ & r · |F_{n}| & ⌉ \\ & & ⌉ \end{aligned}^{∗} < ϵ/2. Use the countable subadditivity and solidity to conclude that ⌈⌈r · ∑ |F_{n}|⌉⌉^{∗} < ϵ. 3.2.15 (i): If F ∈ 𝔏^{1}[\begin{matrix}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{matrix}^{∗}], then there are elementary integrands X_{n} with \begin{aligned}⌈ \\ ⌈ & F − X_{n} & ⌉ \\ & & ⌉ \end{aligned}^{∗} < 2^{−n}. The proof of theorem 3.2.10, suitably adapted, shows that the sequence F_{1} = X_{1}, F_{2} = X_{2} −X_{1}, …, F_{n} = X_{n} −X_{n−1}, … will meet the description. For the converse note that for X_{1}, …, X_{N} ∈ E

⌈⌈F − ∑_{n=1}^{N} X_{n} ⌉⌉^{∗} ≤ ∑_{n=1}^{N} \begin{aligned}⌈ \\ ⌈ & F_{n} − X_{n} & ⌉ \\ & & ⌉ \end{aligned}^{∗} + ∑_{n=N+1}^{∞} \begin{aligned}⌈ \\ ⌈ & F_{n} & ⌉ \\ & & ⌉ \end{aligned}^{∗} .

Given ϵ > 0, find first N so large that the second sum is less than ϵ/2, then find X_{n} ∈ E such that the (finite) first sum is also less than ϵ/2.

(ii) is plain from (i) and the countable subadditivity of the mean.

(iii): Let F_{n} ∈ 𝔏^{1}[\begin{matrix}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{matrix}^{∗}]_{+} with ⌈⌈∑_{n} F_{n} ⌉⌉^{∗} < ∞. Find X_{n} ∈ E_{+} with

\begin{aligned}⌈ \\ ⌈ & F_{n} − X_{n} & ⌉ \\ & & ⌉ \end{aligned}^{∗} ≤ 2^{−n} (why can the X_{n} be chosen to be positive?). Then

\begin{aligned}⌈ \\ ⌈ & ∑_{n} X_{n} & ⌉ \\ & ⌉ \end{aligned}^{∗} ≤ \begin{aligned}⌈ \\ ⌈ & ∑_{n} F_{n} & ⌉ \\ & ⌉ \end{aligned}^{∗} + \begin{matrix}⌈ \\ ⌈ & ∑_{n}(X_{n} − F_{n}) & ⌉ \\ ⌉ \end{matrix}^{∗}

≤ \begin{aligned}⌈ \\ ⌈ & ∑_{n} F_{n} & ⌉ \\ & ⌉ \end{aligned}^{∗} + 2 < ∞ ,

and therefore \begin{aligned}⌈ \\ ⌈ & X_{n} & ⌉ \\ & & ⌉ \end{aligned}^{∗} ⟶_{n→∞} 0. 3.2.16 (i): Let (X_{n}), (Y_{n}) be sequences in E that converge in mean to F, G ∈ 𝔏^{1}[\begin{matrix}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{matrix}^{∗}], respectively, and let r, s ∈ ℝ. Then (rX_{n} + sY_{n}) converges to rF + sG in mean, since

\begin{aligned}⌈ \\ ⌈ & (rF + sG) − (rX_{n} + sY_{n}) & ⌉ \\ & & ⌉ \end{aligned}^{∗} ≤ |r|\begin{matrix}⌈ \\ ⌈ & F − X_{n} & ⌉ \\ & & ⌉ \end{matrix}^{∗} + |s|\begin{matrix}⌈ \\ ⌈ & G − Y_{n} & ⌉ \\ & & ⌉ \end{matrix}^{∗} ,

which converges to zero as n → ∞. Thus

∫ (rF + sG) = lim_{n} ∫ (rX_{n} + sY_{n}) = lim_{n} (r ∫ X_{n} + s ∫ Y_{n})
= r ∫ F + s ∫ G .

This shows that the integral is linear. Next let (X_{n}) be a sequence of ele-

mentary integrands with \begin{aligned}⌈ \\ ⌈ & F − X_{n} & ⌉ \\ & & ⌉ \end{aligned}^{∗} ⟶_{n→∞} 0. Then \begin{aligned}⌈ \\ ⌈ & X_{n} & ⌉ \\ & & ⌉ \end{aligned}^{∗} ⟶_{n→∞} \begin{aligned}⌈ \\ ⌈ & F & ⌉ \\ & & ⌉ \end{aligned}^{∗}

by exercise 3.2.9, and so |∫ F dZ|=lim_{n} |∫ X_{n} dZ|≤lim_{n} \begin{aligned}⌈ \\ ⌈ & X_{n} & ⌉ \\ & ⌉ \end{aligned}^{∗}=\begin{aligned}⌈ \\ ⌈ & F & ⌉ \\ & ⌉ \end{aligned}^{∗}. 3.2.30 If not, one of the collections M_{k} = {A ∩ ((k − 1, k]] : A ∈ M} would contain uncountably many non-negligible sets. For some r ∈ ℕ we would have \begin{aligned}⌈ \\ ⌈ & A & ⌉ \\ & & ⌉ \end{aligned}^{∗} > 1/r for uncountably many A ∈ M_{k}, which is impossible since the measure of ((k − 1, k]] is finite. 3.3.3 The General Stone–Weierstraß theorem A.2.2 permits the extension of the bounded linear map ∫ . dZ : E^{0} → L^{p} to \overline{E}^{0} , so that the A_{t} and E^{0} may be assumed to be both algebras and vector lattices closed under chopping. The argument of lemma 3.3.1, which does not refer to the nature of the elementary integrands, shows that ∫ . Z^{t} is σ-additive in probability. The argument of proposition 3.3.2 also carries through, showing that ∫ . Z is a σ-additive vector measure to L^{p}. Daniell’s mean furnishes an extension satisfying the Dominated Convergence Theorem, and therefore integrating every elementary integrand in E. 3.4.1 There is a sequence of elementary integrands X^{k} with ⌈⌈X^{k} ⌉⌉^{∗} < 2^{−k} for k ≥ 1 and F = ∑_{k≥0} X^{k} a.e. and in mean. Then Y^{K} =^{def} ∑_{k≤K} k|X^{k}| converges to an integrable function G. Set U =^{def} [G > M] ≤ G/M. Then

U = sup_{n,K} 1 ∧ (n·(Y^{K} − (Y^{K} ∧ M))) ∈ E_{+}^{↑} and \begin{aligned}⌈ \\ ⌈ & U & ⌉ \\ & & ⌉ \end{aligned}^{∗} < ϵ for a suitable

choice of M. On U^{c}, ∑_{k} X^{k} converges uniformly to F. 3.4.7 The class of functions ϕ such that ϕ(F_{1}, …, F_{N}) is measurable is closed under pointwise limits, by Egoroff’s theorem, and contains the continuous functions, by theorem 3.4.6. Thus it contains the smallest class of functions closed under pointwise limits and containing the continuous functions, viz. the Borel functions. 3.4.8 Needed 3.5.4 The first statement is clearly true if Z is continuous and adapted. The collection of adapted processes Z such that Z^{T} is predictable is closed under pointwise limits. If the predictable process V has right-continuous paths of finite variation, then V = sup{∑_{i} |V^{q_{i+1}} − V^{q_{i}} |}, where the supremum is extended over all finite rational partitions {0 = q_{0} < q_{1} < q_{2} < …} of [0, ∞). 3.5.5 Let ϵ > 0. There is a K such that ℙ[|f| ≥ K] ≤ ϵ. Let us write

f·((S, T]] · G = G^{(K)} + G′, with G^{(K)} =^{def} f·[|f| < K] · ((S, T]] · G and G′ =^{def}

f·[|f| ≥ K] · ((S, T]] · G = f · ((S_{[|f|≥K]}, T]] · G. G^{(K)} , being Z−0-measurable and majorized by KG, is Z−0-integrable. G′ vanishes off the stochastic interval ((S_{[|f|≥K]}, T]], whose projection on Ω has measure less than ϵ, and thus has ⌈⌈G′ ⌉⌉_{Z−0}^{∗} ≤ ϵ (exercise 3.1.2). That is to say, f·((S, T]] · G differs arbitrarily little (by less than ϵ) from a Z−0-integrable process (G^{(K)} ), so it is Z−0-integrable itself. Furthermore, ∫ f·((S, T]] · G dZ≐lim_{K→∞} ∫ G^{(K)} dZ. It suffices to show that ∫ G^{(K)} dZ ≐f[|f| < K] ∫ ((S, T]] · G dZ; in other words, that equation (3.5.2) holds when f is bounded. If it holds when S and T are elementary, we apply it to the stopping times S^{(n)} ∧ n, T^{(n)} ∧ n and take the limit as n → ∞: we may assume that S, T are elementary. If it holds when f is a set in F_{S}, then it holds for linear combinations of them and their uniform limits: we may assume in addition that f is a set A ∈ F_{S}. In that case the equality in question reads ∫ ((S_{A}, T_{A}]] · G dZ=Ȧ · ∫ ((S, T]] · G dZ. If G ∈ E is an elementary stochastic interval or a linear combination thereof, it is true by inspection; it follows in general by approximation with elementary integrands. 3.5.7 (i): Let (X^{(n)}) be a sequence in P^{ℙ} with pointwise limit X. There are predictable processes X_{ℙ}^{(n)} such that N_{n} =^{def} π_{Ω}[X^{(n)} ≠ X_{ℙ}^{(n)}] is nearly empty. Set X_{ℙ} = lim sup X_{ℙ}^{(n)} . This is a predictable process. The projection of [X ≠ X_{ℙ}] on Ω is contained in the union of the N_{n} and therefore in a negligible set of A_{∞σ}: it is nearly empty itself. In other words, the paths of X and X_{ℙ} agree outside a nearly empty set.

(ii): Let us start by showing that a measurable (see page 23) evanescent process X is predictable. There is a nearly empty set N such that X vanishes off N =^{def} [0, ∞) × N. Now the collection M_{N} of processes Y such that Y · N ∈ P is a monotone class and contains every generator of the measurable processes of the form (s, t]×A, A ∈ F_{∞}^{∗}. Indeed, ((s, t] × A) · N = (s, t] × (A ∩ N) is in E on the grounds that the nearly empty set A ∩ N belongs to F_{0} = F_{0+}^{ℙ}. Therefore M_{N} contains all measurable processes, in particular X. That is to say, X ∈ P. Now if X is a measurable previsible process and X_{ℙ} ∈ P cannot be distinguished from X with ℙ, then X = X_{ℙ} + (X − X_{ℙ}) is the sum of two processes in P. 3.5.10 (i): (t−1/n)∨0 predicts t. (T +(ϵ−1/n) : n > 1/ϵ) predicts T +ϵ. If T^{1}, T^{2}, … are announced by T_{n}^{1}, T_{n}^{2}, …, then T′_{N} = ⋁_{k,n≤N} T_{n}^{k} predicts ⋁_{k} T^{k}, and T_{n}^{1} ∧ … ∧ T_{n}^{k} predicts T^{1} ∧ … ∧ T^{k}. (ii): 0_{A} ∧ n predicts 0_{A}. S_{A} = S ∨ 0_{A}. If (S^{(n)}) announces S, then (S_{[S^{n}≤T]}^{n} ∧ n) announces S_{[S≤T]} . 3.5.11 It suffices to show that [[0, T)) is predictable; all other intervals in question can be gotten from this one by taking intersections and relative complements with intervals then known to be predictable. Let (T_{n}) be a sequence of stopping times announcing T and simply observe that [[0, T)) is a simple combination of predictable sets:

[[0, T)) = ⋃_{n} [[0, T_{n}]] \ [[T]] .

3.5.19 (i) The set {t : ΔV_{t} ≥ λ} is left closed, on the grounds that the non-oscillatory nature of the path of V ∈ 𝔇 prevents it from having an accumulation point. The graph of T_{ΔV}^{λ} is the intersection of the previsible sets [[0, T_{ΔV}^{λ}]] and [ΔV ≥ λ]. The claim follows from theorem 3.5.13. (ii) (See the proof of theorem 2.4.4). For every i, j ∈ ℕ define inductively T^{i,0} = 0 and

T^{i,j+1} = inf {t > T^{i,j} : ΔI_{t} ≥ 1/i} .

From exercise 3.5.19 we know that the T^{i,j} are predictable stopping times. They are countable in number, so we count them: {T^{i,j}} = {T′_{1}, T′_{2}, … }. The T′_{n} do not have disjoint graphs, of course, so we force the issue: since P_{n} =^{def} [[T_{n}]]\⋃_{ν<n}[[T′_{ν}]] is previsible, the reduction T_{n} of T′_{n} to ⋂_{ν<n}[T_{ν} ≠ T_{n}], which has P_{n} for its graph, is a predictable stopping time (see theorem 3.5.13). The T_{n} meet the claim. 3.5.20 Let (S_{n}) be a sequence of stopping times announcing T. By Doob’s optional stopping theorem 2.5.22,

M_{T−} = lim M_{S_{n}} = lim 𝔼[M_{∞}|F_{S_{n}}]

both almost surely and in L^{1}-mean. For every n and A ∈ F_{S_{n}} the integrals of M_{∞} and M_{T−} over A agree. Since F_{T−} = ⋁ F_{S_{n}}, these integrals agree also if taken over any set of F_{T−}.

3.6.8 XF̃ ≥ XF is predictable, so XF̃ ≥ X̃F a.e. By the same argument, X^{−1}X̃F ≥ F̃. Division by zero does not really occur (why?). 3.6.13 Let T^{(n)} be the stopping times of exercise 1.3.20. The equality

∫ X dZ^{T^{(n)}∧k} = ∫ X · [[0, T^{(n)} ∧ k]] dZ

holds trivially for all X ∈ E and all k, n ∈ ℕ. By right-continuity we have

∫ X dZ^{T ∧k} ≐ ∫ X · [[0, T ∧ k]] dZ

for all X ∈ E. Consider the collection of all predictable X for which the equality above obtains. By the Dominated Convergence Theorem this collection is closed under limits of bounded sequences and therefore contains all bounded predictable processes. Letting k → ∞ shows that for all X ∈ P_{00}

∫ X dZ^{T} = ∫ X · [[0, T]] dZ .  (∗)

Now let G̃ be a Z-envelope of |G|. Choose it so that it is also a Z^{T} -envelope of |G|. Then by exercise 3.6.8

\begin{matrix}⌈ \\ ⌈ & G · [[0, T]] & ⌉ \\ ⌉ \end{matrix}_{Z−p}^{∗} = \begin{matrix}⌈ \\ ⌈ & G̃ · [[0, T]] & ⌉ \\ ⌉ \end{matrix}_{Z−p}^{∗}
by corollary 3.6.10:  = sup {⌈⌈∫ Y dZ⌉⌉_{p} : Y ∈ P_{00}, |Y | ≤ G̃ · [[0, T]]}
by (∗):  = sup {⌈⌈∫ Y dZ^{T} ⌉⌉_{p} : Y ∈ P_{00}, |Y | ≤ G̃}

= \begin{matrix}⌈ \\ ⌈ & G & ⌉ \\ ⌉ \end{matrix}_{Z^{T}−p}^{∗} .

The rest is even simpler. Suppose, say, G · [[0, T]] is Z−p-integrable. There is a sequence of elementary integrands X^{(n)} with ⌈⌈G · [[0, T]] − X^{(n)}⌉⌉_{Z−p}^{∗} → 0. Then ⌈⌈G − X^{(n)}⌉⌉_{Z^{T}−p}^{∗} = ⌈⌈(G − X^{(n)}) · [[0, T]]⌉⌉_{Z−p}^{∗} → 0 and ∫ G dZ^{T} = lim ∫ X^{(n)} dZ^{T} =lim ∫ X^{(n)} · [[0, T]] dZ. 3.6.14 The mean \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗} =^{def} \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−0}^{∗} + \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z′−0}^{∗} majorizes d(Z + Z′) on E and therefore \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{(Z+Z′)−0}^{∗} on predictable processes (exercise 3.6.16). Let \underline{F} , \overline{F}

be predictable with \underline{F} ≤ F ≤ \overline{F} and \begin{aligned}⌈ \\ ⌈ & \overline{F} − \underline{F} & ⌉ \\ & & ⌉ \end{aligned}_{Z−0}^{∗} = 0, and let \underline{F}′, \overline{F}′ be similar, constructed with Z′ instead (proposition 3.6.6 (ii)). Set \overline{Φ} =^{def} \overline{F} ∧ \overline{F}′ and \underline{Φ} =^{def} \underline{F} ∧ \underline{F}′. Then \begin{aligned}⌈ \\ ⌈ & \overline{Φ} − \underline{Φ} & ⌉ \\ & & ⌉ \end{aligned}^{∗} = 0 and therefore \begin{aligned}⌈ \\ ⌈ & \overline{Φ} − F & ⌉ \\ & ⌉ \end{aligned}_{(Z+Z′)−0}^{∗} = 0. If F is integrable for both Z and Z′, then \overline{Φ} is finite for \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−0}^{∗} and \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z′−0}^{∗}, therefore for \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗} and for \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{(Z+Z′)−0}^{∗}: it and with it F is (Z + Z′)−0-integrable; etc. 3.6.16 If \begin{aligned}⌈ \\ ⌈ & F & ⌉ \\ & & ⌉ \end{aligned}_{Z−p}^{∗} > a, then ⌈⌈∫ Y dZ ⌉⌉_{L^{p}} > a for some Y as in corollary 3.6.10. Such Y is integrable for any mean; there is a sequence (X^{(n)}) of elementary integrands converging in the mean \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗} + \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−p}^{∗} to Y . Then

\begin{aligned}⌈ \\ ⌈ & F & ⌉ \\ & & ⌉ \end{aligned}^{∗} ≥ \begin{aligned}⌈ \\ ⌈ & Y & ⌉ \\ & & ⌉ \end{aligned}^{∗} = lim_{n} ⌈⌈X^{(n)} ⌉⌉^{∗} ≥ lim_{n} ⌈⌈X^{(n)} ⌉⌉_{Z−p}^{∗} > a.

3.6.18 Start with p ≥ 1. Consider pairs (A, μ_{A}) consisting of a Z-non-negligible predictable Z−p-integrable set A and a positive σ-additive measure μ_{A} that satisfies |μ(X)| ≤ \begin{aligned}⌈ \\ ⌈ & X & ⌉ \\ & & ⌉ \end{aligned}_{Z−p}^{∗} and

μ_{A}(P) = 0 ⟺ \begin{aligned}⌈ \\ ⌈ & P & ⌉ \\ & & ⌉ \end{aligned}_{Z−p}^{∗} = 0  ∀P ∈ P.

A maximal collection of such pairs with mutually disjoint first entries is at most countable, so we write it {(A^{(1)}, μ_{A^{(1)}}), …}. The complement B of ⋃_{k} A_{k} is Z-negligible; if it were not, then the Hahn–Banach theorem A.2.25 would provide a linear functional μ on L^{(1)}[Z−p] with μ(B) > 0. Such μ would be a measure on P, and it could be chosen positive. It is not hard to restrict μ to a subset A^{(0)} of B such that (A^{(0)}, μ_{|A^{(0)}}) could be adjoined to the supposedly maximal collection. μ := ∑_{k} 2^{−k}μ_{A_{k}} meets the description. If 0 ≤ p < 1, then theorem 4.1.2 provides a probability ℙ′ ≈ ℙ for which Z is an L^{2}-integrator. The measure μ produced above in this situation is a control measure for Z. 3.7.2 T^{(1)} ∨ T^{(2)} reduces Z to an L^{p}(ℙ)-integrator. (exercise 2.1.10). We may assume without loss of generality that G is positive. By exercise 3.6.13 the G · ((S^{(i)}, T^{(i)}]] are Z^{T^{(1)}∨T^{(2)}}−p-integrable and then so is their supremum G · ((S^{(1)} ∧ S^{(2)}, T^{(1)} ∨ T^{(2)}]]. 3.7.8 For every ℙ ∈ 𝔓 let ℙ−∫ G dZ denote the stochastic integral computed in L^{0}(ℙ). Let Ξ denote the collection of bounded predictable processes X such that there exists a right-continuous process X∗Z with left limits and with (X∗Z)_{t} ∈ ℙ−∫ X · [[0, t]] dZ for all ℙ ∈ 𝔓 and all t > 0. Clearly E ⊂ Ξ. Let (X^{(n)}) be a bounded sequence in Ξ that converges pointwise to some X ∈ Ξ. By lemma 2.3.2 and exercise 3.6.13

\begin{aligned}⌈ \\ ⌈ & (X^{(n)}∗Z − X^{(m)}∗Z)_{t}^{⋆} & ⌉ \\ & & ⌉ \end{aligned}_{L^{0}(ℙ)} ⟶_{m,n→∞} 0

for every ℙ ∈ 𝔓 and t > 0.^{2} That is to say, the set of points in Ω where the path of X^{(n)}∗Z does not converge uniformly on every finite interval belongs to A_{∞σ} by its description and is ℙ-negligible for every ℙ ∈ 𝔓. We set X∗Z = 0 on this set and X∗Z = lim X^{(n)}∗Z elsewhere. Using the regularity of F. we conclude that X ∈ Ξ: Ξ contains all bounded predictable processes. If X is predictable and locally Z−0;ℙ-integrable for every ℙ ∈ 𝔓, we apply the result to ((−n) ∨ X ∧ n) · [[0, n]] and take the limit. 3.7.9 There are stopping times T_{n} increasing to ∞ that reduce M to global L^{1}-integrators for which G is M^{T_{n}}−1-integrable (corollary 2.5.29). The situation is reduced to the case that M is a martingale and global L^{1}-integrator and G is M−1-integrable. We have to show that then G∗M is

2 Previous faulty formula corrected by Roger Sewell, <rfs@cambridgeconsultants.com>

a martingale, i.e., that 𝔼[∫ X d(G∗M)] = 0 for X ∈ E with X_{0} = 0 (see proposition 2.5.10). This is evident if G is elementary, and follows in general by approximating G with elementary integrands in \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{1−M}^{∗}-mean. 3.7.15 Apply exercise 3.6.13 and lemma 3.7.5: for any t

(G∗Z^{T})_{t} ∈ ∫_{0}^{t} G dZ^{T} = ∫ [[0, T ∧ t]] · G dZ = ∫ [[0, T ∧ t]] d(G∗Z)
∋ (G∗Z)_{T ∧t} = (G∗Z)_{t}^{T} .

Now use the right-continuity of the ultimate right and left-hand sides. 3.7.16 We start with the case that the graph [[T]] is included. Setting

B =^{def} {(s, ω) : ω ∈ Ω_{0}, s ≤ T (ω)} write

X∗Z − X′∗Z′ = (X − X′)∗Z + X′∗(Z − Z′) .  (∗)

It is clear upon inspection that X∗(Z − Z′) = 0 on B if X is an elementary integrand. We know from exercise 3.6.14 that X′ is (Z − Z′)−0-integrable. There is a sequence (X^{(n)}) of elementary integrands such that X^{(n)}∗(Z − Z′) converges to X′∗(Z − Z′) uniformly on bounded intervals, except in a nearly empty set (corollary 3.7.11). We conclude that, on B, X′∗(Z − Z′) is indistinguishable from 0.

To show that the first term of (∗) is evanescent on B let ϵ > 0 and set S = inf {t : ((X − X′)∗Z)_{t}^{⋆} ≥ ϵ}. This is a stopping time (proposition 1.3.11). On Ω_{0} ∩ [S ≤ T] the entire path of (X − X′) · [[0, S]] vanishes. From corollary 3.7.13 in conjunction with lemma 3.7.5 and exercise 3.6.13 we know that for all t

((X − X′)∗Z)_{S∧t} ∈ ∫_{0}^{S∧t}(X − X′) dZ = ∫ (X − X′) · [[0, S]] dZ^{t}

almost surely vanishes on this set. The right-continuity of (X − X′)∗Z shows that the whole path of (X − X′)∗Z vanishes almost surely up to and including time S on Ω_{0} ∩ [S ≤ T]. Since ((X − X′)∗Z)_{S}^{⋆} ≥ ϵ on [S < ∞], this implies that S > T almost surely on Ω_{0} . We take ϵ → 0 and conclude that (X − X′)∗Z vanishes almost surely on Ω_{0} up to and including time T.

If [[T]] is excluded, we apply the above to (T − ϵ) ∨ 0 and let ϵ → 0. 3.7.18 needed 3.7.19 The first statement was done in exercise 3.5.8. For the second statement note that ((S, T]]∗Z = Z^{T} − Z^{S} is previsible if S, T are stopping times. Thus X∗Z is previsible for X ∈ E. Now approximate the general integrand as in corollary 3.7.11. 3.7.20 (i) To say that f · G is Z−0-integrable on ((S, T]] means that f · G · ((S, T]] is Z−0-integrable (exercise 3.6.13). In view of definition 3.7.1 neither hypothesis nor conclusion are changed if we replace G by G · ((S, T]]: we may assume that G vanishes off ((S, T]] and is Z−0-integrable. Then f · G is Z−0-integrable on ((S, T]] if and only if f · ((S, T]] is G∗Z−0-integrable (theorem 3.7.10), which is true because G∗Z is a global L^{0}-integrator and f · ((S, T]] has almost surely finite maximal function at ∞ (theorem 3.7.17). Thus

∫_{S+}^{T} f · G dZ = ∫ f · ((S, T]] d(G∗Z)
by exercise 3.5.5:  = ḟ · ((G∗Z)_{T} − (G∗Z)_{S})˙
by exercise 3.6.13:  = ḟ · ∫ ((S, T]] d(G∗Z)
by theorem 3.7.10:  = ḟ · ∫ G · ((S, T]] dZ

by exercise 3.6.13:  = ḟ · ∫_{S+}^{T} G dZ .

(ii) Again we may assume that G vanishes off the predictable interval [[S, T]] and is Z−0-integrable. Again we know a priori that f · G is Z−0-integrable on [[S, T]]: it amounts to saying that f · [[S, T]] is G∗Z−0-integrable, which is true for the same reason as above. This allows us to reduce the problem to the case that f is bounded. For if we have the equality (3.7.6) in this case, we apply it to (−k) ∨ f ∧ k and let k → ∞. Say |f| ≤ k. Let (S^{(n)}) be an increasing sequence of stopping times announcing S. There is a sequence of F_{S^{(n)}}-measurable functions f^{(n)} with |f^{(n)}| ≤ k that converges almost surely to f; we can take for f^{(n)} the conditional expectation of f given F_{S^{(n)}} , or by density first find the f^{(n)} so as to converge in \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{0}-mean to f and then go to an almost surely convergent subsequence. Now from (i)

∫_{S^{(n)}+}^{T} f^{(m)} · G dZ = ḟ^{(m)} · ∫_{S^{(n)}+}^{T} G dZ ,  m ≤ n ∈ ℕ .

We let first n → ∞ and then m → ∞ and get the claim.

(iii) G is predictable (proposition 3.5.2) and has almost surely finite maximal function at any finite instant t. It is thus Z−0-integrable on every interval [[0, t]] (theorem 3.7.17) with integral ∫ G · [[0, t]] dZ≐∑ f_{k} · (Z_{S_{k+1}}^{t} − Z_{S_{k}}^{t}) (proposition 3.5.2 and theorem 3.2.24). 3.7.21 By corollary A.5.13, |X^{(n)} − X|_{T}^{⋆} is F_{T} -measurable. Thus every subsequence of X^{(n)} has a further subsequence X^{(n_{k})} with |X^{(n_{k})} −X|_{T}^{⋆} ⟶_{k→∞} 0 almost surely. Then X^{(n_{k})} · [[0, T]] → X · [[0, T]] nearly uniformly and thus Z-almost everywhere. By theorem 3.7.17, X^{(n_{k})} · [[0, T]] → X · [[0, T]] in Z−p-mean, by equation (3.7.5) X^{(n_{k})}∗Z^{T} − X∗Z^{T} _{Z−p} ⟶_{k→∞} 0, and the claim follows from theorem 2.3.6. 3.7.24 Make ∥(X_{.−} − X^{n_{k}}) · [[0, k]]∥_{Z−p}^{∗} summable and use Borel–Cantelli.

3.7.25 Let S_{1}, S_{2} be stochastic partitions. Then S =^{def} ⋃{[[S]] : S ∈ S_{1} ∪ S_{2}} is progressively measurable. Define by induction T_{1} = inf{t : t ∈ S} and T_{k+1} =^{def} inf{t > T_{k} : t ∈ S}. T_{1} , being the infimum of the first stopping time of S_{1} and the first stopping time of S_{2} is a stopping time (exercise 1.3.15). If T_{k} is a stopping time, then T_{k+1} is the debut of the progressively measurable set ((T_{1}, ∞)) ∩ S and therefore is a stopping time, by corollary A.5.12. This big result uses the natural conditions, so it is better to argue as follows: [T_{k+1} ≤ t] = ⋃{[T_{k} < S ≤ t] : S ∈ S_{1} ∪ S_{2}}. Now [T_{k} < S] ∈ F_{S} (exercise 1.3.16) and so [T_{k} < S ≤ t] = [T_{k} < S] ∩ [S ≤ t] ∈ F_{t}. The T_{k} and T_{∞} =^{def} sup_{k∈ℕ} T_{k} make up a partition that refines both S_{1} and S_{2} . 3.7.27 Let R =^{def} (X, Z), \underline{R} : Ω → 𝒟^{2} the associated representation so that X = \overline{X}∘\underline{R}, Z = \overline{Z}∘\underline{R}. Now define stopping times \overline{S}_{k} : 𝒟^{2} → \overline{ℝ} and processes \overline{Y} _{t}^{(δ)} on 𝒟^{2} by (3.7.10) and (3.7.11). Choose δ_{n} so that ∑_{n} f(nδ_{n}) < ∞, let and set (x_{.−} ⊕ ∗z). =^{def} lim \overline{Y} .^{(δ_{n})}(x., z.) where this limit exists uniformly on bounded intervals, x_{.−} ⊕ ∗z =^{def} 0 elsewhere. This produces a map: 𝒟^{2} → 𝒟 which is easily seen to be adapted to the canonical (i.e., right-continuous) filtrations on these path spaces. Clearly the version of X_{.−}∗Z produced by the algorithm (3.7.11) is nothing but the composition X_{.−} ⊕ ∗Z of (X, Z) with this map, as long as Z satisfies (3.7.13). 3.7.29 Apply the Borel–Cantelli lemma to this consequence of inequality (3.7.14):

ℙ [|X_{.−}∗Z − Y^{(δ_{n})}|_{∞}^{⋆} > 1/n] ≤ nδ_{n} · Z _{I^{0}} .

3.7.31 (ii) The set [N(U) > K] is the same as [S_{K} < U], and on it

Kδ^{2} ≤ ∑_{k<K} (X_{S_{k+1}}^{U} − X_{S_{k}}^{U})^{2} ≤ L^{2} ∑_{k<K} (X_{S_{k+1}}^{′U} − X_{S_{k}}^{′U})^{2} .

We take the square root and apply corollary 3.1.7, getting

\sqrt{K}δ(ℙ[N > K])^{1/q} ≤ LK_{q}^{(A.8.5)} X^{′U} _{I^{q}} .

Raising this to the power q and using the estimate of K_{q}^{(A.8.5)} from exercise A.8.29 gives the claim. 3.7.32 (i) The approximands of corollary 3.7.11 have continuous paths.

(ii) Let (X^{(n)}) be a sequence of elementary integrands approximating X in mean, fast enough so that X^{(n)}∗Z → X∗Z uniformly on compacta a.s. (corollary 3.7.11). Since the equation in question holds by inspection for elementary integrands,

Δ(X∗Z) = lim Δ(X^{(n)}∗Z) = lim X^{(n)} · ΔZ = X · ΔZ .

(iii) The same argument works if X^{(n)} is chosen as in theorem 3.7.26. 3.7.35 Use theorem A.3.18 on page 403.

3.8.3 Convolving the absolute value function |.| with a suitable approximate identity δ that is of class C^{∞}, has compact support, and is symmetric about the origin produces C^{∞}-functions |.|^{(δ)} that converge uniformly on every compact set in ℝ^{d} to |.|; also, the gradients of the |.|^{(δ)} will converge pointwise and dominatedly on every compact to ∇_{η}|.|. Applying theorem 3.8.1 to the |.|^{(δ)} and taking the limit will result in the claim. 3.8.6 Apply exercise 3.7.19. 3.8.8 For η = 1, …, d let S = {0 = S_{0} ≤ S_{1} ≤ S_{2} ≤ … ≤ ∞} be a random partition^{8}, and set f_{0}^{η} = Z_{0}^{η}, f_{1}^{η} = Z_{S_{1}}^{η} − Z_{0}^{η}, f_{k}^{η} = Z_{S_{k}}^{η} − Z_{S_{k−1}}^{η}, k = 2, 3 …, K. Enumerate the f ’s: f_{1}, … f_{(K+1)h}. Let ϵ_{ν} be the Bernoulli random variables of theorem A.8.26. We have

( ∑_{ν=1}^{(K+1)d} f_{ν}^{2})^{1/2} ≤ K_{p}^{(A.8.5)} · ∥ ∑_{ν=1}^{(K+1)d} f_{ν}ϵ_{ν} ∥_{L^{p}(τ)}

and so ∥( ∑_{ν=1}^{(K+1)d} f_{ν}^{2})^{1/2}∥_{L^{p}(ℙ)} ≤ K_{p} · ∥∥ ∑_{ν=1}^{(K+1)d} f_{ν}ϵ_{ν} ∥_{L^{p}(τ)}∥_{L^{p}(ℙ)}

= K_{p} · ∥∥ ∑_{ν=1}^{(K+1)d} f_{ν}ϵ_{ν} ∥_{L^{p}(ℙ)}∥_{L^{p}(τ)}

≤ K_{p} · Z^{T} _{I^{p}[ℙ]} .

This is because the sum in the penultimate line is of the form ∫_{0}^{T} ⟨X|dZ⟩ with X ∈ E_{1}^{d}. As the partition S runs through a sequence whose mesh tends to zero, the left-hand side of this inequality converges to (∑_{η=1}^{d}[Z^{η}, Z^{η}]_{T} )^{1/2} . The case p = 0 is handled similarly (see the proof of corollary 3.1.7).

3.8.11 Thanks to Roger Sewell, <rfs@cambridgeconsultants.com>, who

saw that the claim needed additional hypotheses and the right constant to become true.

There are arbitrarily large stopping times T_{1} such that M_{.−}^{T_{1}} and N_{.−}^{T_{1}} are bounded. For instance, the infimum of

inf{t : |M_{t}| > k} and inf{t : |N_{t}| > k}

tends to ∞ as k → ∞. There are arbitrarily large stopping times T_{2} such that both M^{T_{2}} and N^{T_{2}} are L^{2}-integrators. T =^{def} T_{1} ∧ T_{2} can be made arbitrarily large. The first two terms on the right in

M_{T} ·N_{T} = ∫_{0+}^{T} M_{.−} dN + ∫_{0+}^{T} N_{.−} dM + [M, N]_{T}

are the values at T of martingales that vanish at t = 0. Their expectation is zero. The last claim follows from theorem 2.5.19.

Suppose M is merely a càdlàg local martingale. There are arbitrarily large stopping times T such that both M_{.−}^{T} is bounded and M^{T} an L^{1}-integrator (corollary 2.5.29). In the formula

M_{T}^{2} = 2 ∫_{0+}^{T} M_{.−} dM + [M, M]_{T}

either side is integrable iff (Δ_{T} M)^{2} is, and if it is not then both sides have expectation ∞. Thus

𝔼[M_{T}^{2}] = 𝔼[[M, M]_{T} ] and 𝔼[M_{T}^{⋆2}] ≤ 4 · 𝔼[[M, M]_{T} ]

for arbitrarily large stopping times T.

3.8.12 Given ϵ > 0 set S_{0} =^{def} 0 and S_{k+1} =^{def} T ∧inf{t > S_{k} : |V_{t}^{c}−V_{S_{k}}^{c}| ≥ ϵ}.

For this partition^{8}

A_{T}^{T} [ .^{2}; V^{c}] ≤ ϵ ∑_{n} |V_{S_{k+1}} − V_{S_{k}}| ≤ ϵ · V _{T} .

Hence S[^{c}V] = 0 and S[V] = S[^{j}V +^{j}V] ≤ S[^{j}V] = S[V −^{c}V] ≤ S[V]. The second claim follows from the inequality of Kunita–Watanabe.

  Thanks to Roger Sewell, <rfs@cambridgeconsultants.com> for the fol-

lowing emphasis: The last equality makes sense only if ∫ Z dV is understood as an ordinary pathwise Lebesgue-Stieltjes integral (see page 144). Then

∫_{0}^{t} Z dV = ∫_{0}^{t} Z_{.−} dV + ∫_{0}^{t} ΔZ dV = ∫_{0}^{t} Z_{.−} dV + ∑_{0≤s<t} ΔZ_{s}ΔV_{s} =

∫_{0}^{t} Z_{.−} dV +[Z, V]_{t}, and the stated equality follows from the definition (3.8.8) of the square bracket. [By proposition 3.7.33 on page 144, ∫_{0}^{t} Z_{.−} dV can be understood as a Lebesgue Stieltjes integral or a stochastic integral; either way its value is the same.] 3.8.13 Without loss of generality M_{0} = 0. Since M is continuous and has finite variation, [M, M] = 0 (exercise 3.8.12). There are thus arbitrarily large times T with 𝔼[M_{T}^{2}] = 0 (exercise 3.8.11). By theorem 2.5.19 there are arbitrarily large bounded stopping times T with 𝔼[M_{T}^{⋆2}] = 0: the set [M_{.−}^{⋆} ≠ 0] is evanescent. 3.8.14

Y_{s}Z_{s} = ∫_{0+}^{s} Y_{.−} dZ + ∫_{0+}^{s} Z_{.−} dY + [Y, Z]_{s}

= Y_{0}Z_{0} + ∑_{0≤k<∞} (Y^{S_{k+1}}Z^{S_{k+1}} − Y^{S_{k}}Z^{S_{k}})_{s}

= Y_{0}Z_{0} + ∑_{0≤k<∞} (Y_{s}^{S_{k+1}} − Y_{s}^{S_{k}})(Z_{s}^{S_{k+1}} − Z_{s}^{S_{k}})

+ ∑_{0≤k<∞} Y_{S_{k}}(Z_{s}^{S_{k+1}} − Z_{s}^{S_{k}}) + ∑_{0≤k<∞} Z_{S_{k}}(Y_{s}^{S_{k+1}} − Y_{s}^{S_{k}})

= Y_{0}Z_{0} + ∑_{0≤k<∞} (Y_{s}^{S_{k+1}} − Y_{s}^{S_{k}})(Z_{s}^{S_{k+1}} − Z_{s}^{S_{k}})

+ ∫_{0+}^{s} Y_{.−}^{S} dZ + ∫_{0+}^{s} Z_{.−}^{S} dY

Subtract the first line from the last and rearrange to obtain

[Y, Z]_{s} − (Y_{0}Z_{0} + ∑_{0≤k<∞}(Y_{s}^{S_{k+1}} − Y_{s}^{S_{k}})(Z_{s}^{S_{k+1}} − Z_{s}^{S_{k}}))

= ∫_{0}^{s}(Y^{S} − Y)_{.−} dZ + ∫_{0}^{s}(Z^{S} − Z)_{.−} dY .

Now apply theorem 2.3.6 and theorem 3.7.10. 3.8.24 Roger Sewell, <rfs@cambridgeconsultants.com>, points out that this is put way too cavalierly: “is constant” should be read “has a nearly constant modification,” and even this statement needs the natural conditions. Namely, only when the filtration is right continous do proposition 2.5.13 on page 75 or the finite variation furnish an adapted càdlàg modification to which the following arguments apply.

(i) Let Z be a previsible local martingale of finite variation, which in vieew of the natural conditions may be assumed càdlàg. Set M =^{def} Z − Z_{0} . 2.5.29 and 3.5.16 provide arbitrarily large stopping times T such that M^{T} is a bounded global L^{1}-integrator. Since M has finite variation, the continuous bracket {M, M} vanishes (exercise 3.8.12) and proposition 3.8.22 results in

M_{T}^{2} = ∫ (M + M_{.−}) dM^{T} .

Since M is a martingale, the right-hand side has expectation zero (exercise 3.2.17). By Doob’s maximal theorem 2.5.19, 𝔼[M_{T}^{⋆2} ] = 0 for arbitrarily large stopping times T: M is indeed evanescent, and Z = M_{0} is nearly constant. (ii) Again 2.5.29 and 3.5.16 provide arbitrarily large stopping times T such that M^{T} is a global L^{1}-integrator and V ^{T} is bounded. By exercise 3.8.12

[M, V]_{S}^{T} − [M, V]_{0}^{T} = ∑_{0<s≤T ∧S} ΔM_{s}·ΔV_{s} = ∫_{0+}^{S} ΔV dM^{T}

for any stopping time S; since this quantity has expectation zero, [M, V]^{T} is by proposition 2.5.10 a martingale, and [M, V] is a local martingale. 3.8.25 (i) Linearity and the Dominated Convergence Theorem reduce the situation to the case that f is positive and bounded. By proposition 3.6.6 (ii) there is a positive previsible process P ≤ f·[[T]] that differs Z−0-negligibly from f·[[T]]. By theorem 3.5.13, [P > 0] is the graph of a predictable stopping time S, announced by the sequence (S_{n}), say. By lemma 3.5.15, P_{S} is measurable on the strict past F_{S−} and can be approximated in measure by random variables f_{n} ∈ F_{S_{n}}. Taking a subsequence, we can arrange things so that f_{n} → P_{S} almost surely. Then f_{n}·((S_{n}, T]] → P = P_{S}·[[S]] except on an evanescent set. Taking the limit in

∫ f_{n}·((S_{n}, S]] dZ = f_{n} · (Z_{S} − Z_{S_{n}})

results in

∫ f·[[T]] dZ = ∫ P dZ = P_{S} · ΔZ_{S} .

Now since the L^{0}-integrators f·[[T]]∗Z and P ∗Z are the same and have the same jumps, and since the jump P_{S}·ΔZ_{S} of the latter vanishes almost surely on [T < S] = [S = ∞], the jump f·ΔZ_{T} of the former also vanishes almost surely on this set, and the consequent equality P_{S}·ΔZ_{S} = f·ΔZ_{T} results in the claim. (ii) In the proof of proposition 3.8.22 apply (i) instead of theorem 3.5.13 and continue as in the proof of proposition 3.8.22. 3.9.1 For any F ∈ C^{2}(D) set \overline{F} =^{def} F ∘ Z, so that \overline{Φ} = Φ(Z) and \overline{Φ_{;η}} = Φ_{;η}(Z), etc. Let Φ, Ψ ∈ I. That is to say,

d\overline{Φ}_{t} = \overline{Φ_{;η}}_{t−}dZ_{t}^{η} + \frac{1}{2}\overline{Φ_{;ηθ}}_{t}d[Z^{η}, Z^{θ}]_{t}^{c} + (Δ\overline{Φ}_{t}−\overline{Φ_{;η}}_{t−}ΔZ_{t}^{η})
and  d\overline{Ψ}_{t} = \overline{Ψ_{;η}}_{t−}dZ_{t}^{η} + \frac{1}{2}\overline{Ψ_{;ηθ}}_{t}d[Z^{η}, Z^{θ}]_{t}^{c} + (Δ\overline{Ψ}_{t}−\overline{Ψ_{;η}}_{t−}ΔZ_{t}^{η}) .
Hence d[\overline{Φ}, \overline{Ψ}]_{t} = d[\overline{Φ}, \overline{Ψ}]_{t}^{c} + Δ\overline{Φ}_{t}Δ\overline{Ψ}_{t}
= \overline{Φ_{;η}}_{t}\overline{Ψ_{;θ}}_{t}d[Z^{η}, Z^{θ}]_{t}^{c} + Δ\overline{Φ}_{t}Δ\overline{Ψ}_{t}
and  d(\overline{Φ} · \overline{Ψ})_{t} = \overline{Φ}_{t−}d\overline{Ψ}_{t} + \overline{Ψ}_{t−}d\overline{Φ}_{t} + d[\overline{Φ}, \overline{Ψ}]_{t}
= (\overline{Ψ} · \overline{Φ_{;η}} + \overline{Φ} · \overline{Ψ_{;η}})_{t−}dZ_{t}^{η}  (∗)
+ \frac{1}{2}(\overline{Ψ} · \overline{Φ_{;ηθ}} + \overline{Φ} · \overline{Ψ_{;ηθ}} + 2\overline{Φ_{;η}} · \overline{Ψ_{;θ}})_{t}d[Z^{η}, Z^{θ}]_{t}^{c}  (∗∗)
+ \overline{Ψ}_{t−}(Δ\overline{Φ}_{t}−\overline{Φ_{;η}}_{t−}ΔZ_{t}^{η}) + \overline{Φ}_{t−}(Δ\overline{Ψ}_{t}−\overline{Ψ_{;η}}_{t−}ΔZ_{t}^{η}) (∗∗∗)
+ Δ\overline{Φ}_{t}Δ\overline{Ψ}_{t} .  (∗∗∗∗)
Now  Δ\overline{ΦΨ}_{t} = \overline{Ψ}_{t−}Δ\overline{Φ}_{t} + \overline{Φ}_{t−}Δ\overline{Ψ}_{t} + Δ\overline{Φ}_{t}Δ\overline{Ψ}_{t}
and\overline{(ΦΨ)_{;η}}_{t−}ΔZ_{t}^{η} = (\overline{Φ} · \overline{Ψ_{;η}} + \overline{Ψ} · \overline{Φ_{;η}})_{t−}ΔZ_{t}^{η} .

We see that the items in lines (∗∗∗) and (∗∗∗∗) add up to Δ\overline{ΦΨ}_{t} − \overline{ΦΨ_{;η}}_{t−}ΔZ_{t}^{η}. Identifying the products in lines (∗) and (∗∗) by means of Leibniz’ product rule gives

d\overline{ΦΨ}_{t} = \overline{(ΦΨ)_{;η}}_{t−}dZ^{η} + \frac{1}{2}\overline{(ΦΨ)_{;ηθ}}_{t}d[Z^{η}, Z^{θ}]_{t}^{c} + Δ\overline{ΦΨ}_{t} − \overline{(ΦΨ)_{;η}}_{t−}ΔZ_{t}^{η} .

This says that ΦΨ ∈ I as claimed. 3.9.3 Both existence and uniqueness are immediate from theorem 5.2.15 on page 291. It is also easy to see that a solution ℰ must equal the righthand side ′ℰ of (3.9.4): by exercise 3.7.16, ℰ satisfies ℰ = 1 + ℰ_{.−}∗Z^{T_{1}−} on [[0, T_{1})), where T_{1} = inf{t : ℰ_{t} ≤ 0}. Applying Itô’s Formula with Φ = ln shows that ℰ = ′ℰ on [[0, T_{1})). By proposition 3.8.21, ℰ = ′ℰ on [[0, T_{1}]]. Then continue as in the existence proof. For the second claim, compute: d(ℰ [Z]ℰ [Z′]) = ℰ [Z]_{.−}dℰ [Z′] + ℰ [Z′]_{.−}dℰ [Z] + d[ℰ [Z], ℰ [Z′]] = (ℰ [Z]ℰ [Z′])_{.−}d(Z + Z′ + d[Z, Z′]). 3.9.4 (ii): t ↦ e^{Z_{t}−Z_{0}−^{c}[Z,Z]_{t}/2} is clearly bounded away from zero for 0 ≤ t < S ∧ u so the question is whether the product

∏_{0<s<S∧t}(1 + ΔZ_{s})e^{−ΔZ_{s}} = ∏_{0<s<S∧t,|ΔZ_{s}|<1/2} (1 + ΔZ_{s})e^{−ΔZ_{s}} · ∏_{0<s<S∧t,|ΔZ_{s}|≥1/2} (1 + ΔZ_{s})e^{−ΔZ_{s}}

is bounded away from zero for such t. Now between 0 and S ∧ u there are only finitely many jumps of size |ΔZ_{s}(ω)| ≥ 1/2, by assumption none of them equal to −1, so the contribution of the second product on the right is bounded away from zero. Since ln(1 + u) − u ≥ −2u^{2} for |u| < 1/2, the contribution of the first product exceeds e^{−2^{j}[Z,Z]_{u}(ω)} . 3.9.8 Since [M, M]_{∞} = ∞, the T^{λ+} are almost surely finite. The time transformation λ ↦ T^{λ} is left-continuous, λ ↦ T^{λ+} is right-continuous (theorem 2.4.7) and so is G. (exercise 1.3.30 (iii)). Due to theorem 2.5.22, (W_{λ}, G_{λ}) is a right-continuous local martingale. Let 0 ≤ λ < μ < ∞ and A ∈ G_{λ}, and set X =^{def} A · (λ, μ] ∈ P[G].

Since  λ < [M, M]_{τ} ≤ μ ⟺ T^{λ+} < τ ≤ T^{μ+} ,
we have X_{[M,M]} = A · ((T^{λ+}, T^{μ+}]] ∈ P[F]
and ∫ X dW = A · (W_{μ} − W_{λ}) = A · (M_{T^{μ+}} − M_{T^{λ+}}) = ∫ X_{[M,M]} dM .

That is to say, the vector space and bounded monotone class of processes X ∈ P[G] such that X_{[M,M]} ∈ P[F] and ∫ X dW = ∫ X_{[M,M]} dM contains a linear generator of P[G] and thus contains all bounded G-predictable processes (theorem A.3.4). The usual truncation argument does the rest (use corollary 3.6.10).

To prove that W. is a standard Wiener process we will show that [W, W]_{μ} = μ and invoke corollary 3.9.5. First, for μ > 0,

[W, W]_{μ} = W_{μ}^{2} − 2 ∫_{0}^{μ} W_{λ−} dW_{λ}

= M_{T^{μ+}}^{2} − 2 ∫_{0}^{T^{μ+}} W_{[M,M]_{s}−} dM_{s}

= [M, M]_{T^{μ+}} + 2 ∫_{0}^{T^{μ+}} (M_{s} − W_{[M,M]_{s}}) dM_{s}

= μ + 2 ∫_{0}^{T^{μ+}} (M_{s} − W_{[M,M]_{s}−}) dM_{s} .  (∗)

The last integral is the value at T^{μ+} of a continuous local martingale whose square at T^{μ+} has expectation

𝔼[∫_{0}^{T^{μ+}} (M_{s} − W_{[M,M]_{s}−})^{2} d[M, M]_{s}]
by 2.4.7:  =𝔼[∫_{0}^{μ} (M_{T^{λ+}} − W_{[M,M]_{T^{λ+}} −})^{2} dλ]
=𝔼[∫_{0}^{μ} (M_{T^{λ+}} − M_{T^{λ}})^{2} dλ]
= ∫_{0}^{μ} 𝔼[(M_{T^{λ+}} − M_{T^{λ}})^{2}] dλ = 0 ,

because M is almost surely constant on [[T^{λ}, T^{λ+}]]. Thus [W, W]_{μ} = μ for all μ, and W is a standard Wiener process. Next let X be left-continuous and adapted to F. . Then X_{T.} is leftcontinuous and adapted to G. . The monotone class theorem shows that X_{T.} is predictable on G. for all X ∈ P[F.]. Let X be the elementary integrand A · (s, t], A ∈ F_{s}. To see that ∫ X dM = ∫ X_{T^{λ}} dW_{λ} it suffices by the above to show that ∫ X_{T^{[M,M]}} dM = ∫ X dM. Now

X_{T^{[M,M]}} = A ∩ [s < T^{[M,M]} ≤ t] = A ∩ [[M, M]_{s} < [M, M] ≤ [M, M]_{t}]
= A ∩ {σ : T^{[M,M]_{s}+} < σ ≤ T^{[M,M]_{t}+}}
so that Δ =^{def} |X − X_{T^{[M,M]}} | = A ∩ (((s, T^{[M,M]_{s}+}]] ∪ ((t, T^{[M,M]_{t}+}]]) .

T^{[M,M]_{s}+} , being the first time [M, M] is strictly greater than [M, M]_{s}, is a stopping time. As [M, M] is constant on Δ, [Δ∗M, Δ∗M] = Δ^{2}∗[M, M] = 0, and we see as above that, indeed, ∫ X dM = ∫ X_{T^{[M,M]}} dM and therefore

∫ X_{t} dM_{t} = ∫ X_{T^{λ}} dW_{λ} .  (∗)

The monotone class of processes for which this equality holds contains thus a generator of P[F.], and then every bounded predictable process. Let now 0 ≤ p < ∞ and let X. be M−p-integrable. In showing that X_{T.} is W−p-integrable and satisfies (∗), we may without loss of generality assume that X ≥ 0. Proposition 3.6.6 on page 125 (ii) provides upper and lower envelopes \begin{matrix}X \\ \end{matrix}∈ P[F.] and X̃ ∈ P[F.] that differ M-negligibly, are finite for \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{M−p}^{∗}, and sandwich X: 0 ≤ \begin{matrix}X \\ \end{matrix}≤ X ≤ X̃. Clearly \begin{matrix}X \\ \end{matrix}_{T.} ≤ X_{T.} ≤ X̃_{T.} . Several applications of regularity (corollary 3.6.10 on page 128) now show that the lower and upper P[G.]-envelopes \begin{matrix}X \\ \end{matrix}_{T.} and X̃_{T.} of X_{T.} differ W -negligibly, that \begin{aligned}⌈ \\ ⌈ & X. & ⌉ \\ & & ⌉ \end{aligned}_{M−p}^{∗} = ⌈⌈X_{T.} ⌉⌉_{W−p}^{∗}, and that (∗) obtains. Thanks to Roger Sewell, <rfs@cambridgeconsultants.com>, who pointed out that the solution previously given here was garbled and beset with typos. 3.9.12 (i) Let T be a stopping time at which G^{′T} and M^{T} are bounded and therefore are martingale L^{2}-integrators. Then

(G_{t}^{′T} )^{2} = 1 + 2 (G^{′T} ∗G^{′T})_{t} + ∫_{0}^{t} (G_{s}^{′T} )^{2} d[M, M]_{s}^{T}

≤ 1 + 2 (G^{′T} ∗G^{′T})_{t} + ∫_{0}^{t} (G_{s}^{′T} )^{2}η_{s}^{2} ds

and so 𝔼[(G_{t}^{′T} )^{2}] ≤ 1 + 𝔼[∫_{0}^{t}(G_{s}^{′T})^{2}η_{s}^{2} ds] = 1 + ∫_{0}^{t} 𝔼[(G_{s}^{′T})^{2}]η_{s}^{2} ds .

Gronwall’s lemma A.2.35 gives 𝔼[(G′_{t∧T} )^{2} ] ≤ exp(∫_{0}^{t} η_{s}^{2} ds). Now take T ↑ ∞ and apply Fatou’s lemma to arrive at 𝔼[G_{t}^{′2} ] ≤ exp(∫_{0}^{t} η_{s}^{2} ds). For (ii) see [82] and [55, page 199]. For the last claim consult [66]. 3.9.15 There is an increasing sequence of stopping times T_{n} ∈ 𝔗 whose pointwise limit is ∞ ℙ-almost surely. Let 0 < t < ∞ and set N =^{def} ⋂_{n}[T_{n} ≤ t] . This is a ℙ-nearly empty set and belongs to F_{T_{n}} ∩ F_{t} since F. is regular. The σ-additivity of ℙ′ gives

𝔼[G′_{t} ] ≥ 𝔼[G′_{t}[T_{n} > t]] = ℙ′[[T_{n} > t] ∪ N] → ℙ′[Ω] = 1 .

3.9.18 Let Ω′ =^{def} Ω\N, and F′_{t} = {A∩Ω′ : A ∈ F_{t}} the filtration induced on Ω′. It is naturally measured by the collection 𝔓′ of probabilities ℙ′ : A∩Ω′ ↦ ∫_{A} dℙ, ℙ ∈ 𝔓. Let then (F′_{t}, \overline{ℙ}′_{t}) be a consistent system of σ-additive

probabilities with \overline{ℙ}′_{t} ≪ ℙ′t on F′_{t}, ℙ′ ∈ 𝔓′, t ≥ 0. Let \overline{ℙ}′ : A′_{∞} =^{def} ⋃_{t} F′_{t} → ℝ

be its projective limit. Again it is to be shown that if A′_{∞} ∋ A′_{n} ↓ ∅, then \overline{ℙ}′[A′_{n}] → 0. We write A′_{n} = A_{n} ∩ Ω′, where A_{n} ∈ A_{∞} can be chosen to decrease as n increases. Now F_{t} ∋ A ↦ \overline{ℙ}_{t}[A] =^{def} \overline{ℙ}′_{t}[A ∩ Ω′], t ≥ 0, clearly defines a consistent system (F., \overline{ℙ}.) such that \overline{ℙ}_{t} ≪ ℙ_{t} on F_{t}, ℙ ∈ 𝔓, t ≥ 0. For if ℙ ∈ 𝔓 vanishes on A ∈ F_{t}, then ℙ′[A ∩ Ω′] = 0 and thus \overline{ℙ}_{t}[A] = \overline{ℙ}′[A ∩ Ω′] = 0. Let \overline{ℙ} denote its σ-additive projective limit on F_{∞}. Then lim \overline{ℙ}′[A′_{n}] = lim \overline{ℙ}[A_{n}] = \overline{ℙ}[⋂ A_{n}] ≤ \overline{ℙ}[N] = 0 (see equation (3.9.8) on page 167). 3.9.23 X · Z = X∗Z + Z∗X +^{c}[X, Z] = (X∗Z +^{c}[X, Z]/2) + (Z∗X +^{c}[X, Z]/2)=X∘Z + Z∘X. 3.10.4 This is evident if F̌ is a linear combination of functions in H. An arbitrary function in E[H] is the confined uniform limit of such and its indefinite integral is therefore uniformly on bounded intervals the limit of the indefinite integrals of such, at least nearly (see inequality (3.10.3)). Then clearly X̌∗ζ is F.[ζ]-adapted, and another application of inequality (3.10.3) gives the claim. 4.1.3 An application of Hölder’s inequality with conjugate exponents q/p and q/(q − p) yields

∫ |f|^{r} dμ = ∫ g · |f|^{r} dμ/g ≤ (∫ g^{q/(q−p)} dμ/g)^{(q−p)/p}·(∫ |f|^{rq/p}dμ/g)^{p/q}
≤ c · ∥f∥_{L^{rq/p}(μ/g)}^{r} .

(ii) is the first half of exercise A.8.17. (iii) Let X ∈ 𝔏^{1}[Z−q; ℙ′]. There are elementary integrands X^{(n)} converging in 𝔏^{1}[Z−q; ℙ′] to X. Then

Δ^{(n)} =^{def} X∗Z − X^{(n)}∗Z has ∥|Y ∗Δ^{(n)}|_{∞}^{⋆} ∥_{L^{q}(ℙ′)} → 0 uniformly for Y ∈ P

with |Y | ≤ 1. Then, with r = p in (i), sup_{Y} ∥|Y ∗Δ^{(n)}|_{∞}^{⋆} ∥_{L^{p}(ℙ)} → 0, which implies Δ^{(n)} _{I^{p}[ℙ]} → 0 and X^{(n)} → X in 𝔏^{1}[Z−p; ℙ]. 4.1.5 According to criterion 4.1.4 there is a strictly positive function g_{0} with ∥g_{0} ∥_{L^{p/(q−p)}(ℙ)} ≤ 1 such that for every x ∈ E

(∫ |I(x)|^{q} · \frac{dℙ}{g_{0}} )^{1/q} ≤ η_{p,q}(I) · ∥x∥_{E} ,  x ∈ E .

What is left to do is to adjust g_{0} so that \frac{dℙ}{g_{0}} becomes a probability.

Now since  ∫ \frac{dℙ}{g_{0} + 0} = ∫ (g_{0}^{p/(q−p)})^{(p−q)/p} dℙ

by theorem A.3.24:  > (∫ g_{0}^{p/(q−p)} dℙ)^{(p−q)/p} ≥ 1  (∗)
and  ∫ \frac{dℙ}{g_{0} + 1} < 1 ,

there is a number σ ∈ (0, 1) such that the function g =^{def} g_{0} + σ satisfies ∫ g^{−1}dℙ = 1. Then g′ =^{def} g^{−1} is bounded and ℙ′ = g′ℙ is a probability. (If the inequality on the left in (∗) is not strict, we face by exercise A.3.27 (iii) the case of a constant g_{0} ; it is silly to contemplate this case, and anyway then g′ = 1/g_{0} = 1 is a priori bounded.) The remaining inequalities are easy to establish: by exercise A.8.2,

∥g ∥_{L^{p/(q−p)}(ℙ)} ≤ 2^{((1− \frac{p}{q−p})/\frac{p}{q−p}) ∨0} · (1 + σ) ≤ 2^{(p∨(q−p))/p} ,

which is inequality (4.1.14). The estimate (4.1.15) follows from exercise 4.1.3. [Thanks to Roger Sewell, <rfs@cambridgeconsultants.com>, who spotted a couple of typos in the first version (12/21/2006).] 4.1.6 (i): Let 0 < p < q < \overline{q} and suppose \overline{C} =^{def} η_{p,\overline{q}}(I) is finite. Then there is a \overline{g} ≥ 0 with ∫ \overline{g}^{p/(\overline{q}−p)} dμ ≤ 1 and ∫ |I(x)|\overline{q}/\overline{g} dμ ≤ \overline{C}\overline{q}∥x∥\overline{q}.

Set g =^{def} \overline{g}^{(q−p)/(\overline{q}−p)} = \overline{g}^{q/\overline{q}} · \overline{g}^{p(q−\overline{q})/\overline{q}(\overline{q}−p)} . Then ∫ g^{p/(q−p)} dμ ≤ 1, and

Hölder’s inequality gives ∫ |I(x)|^{q}/g dμ ≤ \overline{C}^{q}∥x∥^{q}. (ii): Let x_{1}, …, x_{n} ∈ E, let C > η_{p,q}(I) and C′ > η_{p,q}(I′), and set ξ =^{def} (Ix_{1}, …, Ix_{n}) and ξ′ =^{def} (I′x_{1}, …, I′x_{n}). Then by exercise A.8.2

∥∥ξ + ξ′∥_{ℓ^{q}} ∥_{L^{p}(μ)} ≤ 2^{0∨[(1−q)/q]} × ∥∥ξ∥_{ℓ^{q}} + ∥ξ′∥_{ℓ^{q}} ∥_{L^{p}(μ)}

≤ 2^{0∨[(1−q)/q]} · 2^{0∨[(1−p)/p]} × [∥∥ξ∥_{ℓ^{q}} ∥_{L^{p}(μ)} + ∥∥ξ′∥_{ℓ^{q}} ∥_{L^{p}(μ)}]

≤ 2^{0∨[(1/q−1)]} · 2^{0∨[(1/p−1)]} × [C + C′](∑_{ν=1}^{n} ∥x_{ν}∥_{E}^{q})^{1/q} .

That is to say,

η_{p,q}(I + I′) ≤ 2^{0∨(1−q)/q} · 2^{0∨(1−p)/p} × [η_{p,q}(I) + η_{p,q}(I′)] .

Page 195.. It suffices to prove inequality (4.1.13) for a step function

f = ∑_{i} x_{i} A_{i} ,  x_{i} ∈ E , A_{i} ∈ T .

Then

∥∥If∥_{L^{q}(τ)}∥_{L^{p}(μ)} = ∥∥∑_{i} Ix_{i} A_{i}∥_{L^{q}(τ)}∥_{L^{p}(μ)}

= ∥(∑_{i} |Ix_{i}|^{q}τ(A_{i}))^{1/q}∥_{L^{p}(μ)}

≤ C(∑_{i} ∥x_{i}∥_{E}^{q} τ(A_{i}))^{1/q} = C ∥f∥_{L^{q}(τ,E)} .

4.1.8 Such I can be extended via the methods of chapter 3 to a continuous linear map of the bounded Baire functions on K to L^{p}(μ), preserving the modulus of continuity. For the second claim use the first one on the Gelfand transform Î (see page 370). (4.1.18) Indeed, let {x_{ν}}_{ν=1}^{n} be a finite collection of vectors in ℓ^{∞}(k). x_{ν} has an expansion x_{ν} = ∑_{κ} e_{κ} · x_{ν}^{κ} in terms of the standard basis {e_{κ}}_{κ=1}^{k} of ℓ^{∞}(k), and

|Ix_{ν} |^{2} = (∑_{κ=1}^{k} Ie_{κ} · x_{ν}^{κ} )^{2} ≤ ∑_{κ} (Ie_{κ})^{2} ∑_{κ} (x_{ν}^{κ})^{2} ≤ k ∑_{κ} (Ie_{κ})^{2}∥x_{ν} ∥_{ℓ^{∞}}^{2} .

Therefore

∥(∑_{ν} |Ix_{ν}|^{2})^{\frac{1}{2}} ∥_{L^{p}(μ)} ≤ \sqrt{k} · ∥(∑_{κ} (Ie_{κ})^{2})^{\frac{1}{2}} ∥_{L^{p}(μ)} · (∑_{ν} ∥x_{ν}∥_{ℓ^{∞}(k)}^{2})^{\frac{1}{2}}

≤ \sqrt{k} · ∥(∑_{κ} (Ie_{κ})^{p})^{\frac{1}{p}} ∥_{L^{p}(μ)} · (∑_{ν} ∥x_{ν}∥_{ℓ^{∞}(k)}^{2})^{\frac{1}{2}}

= \sqrt{k} · (∑_{κ} ∥Ie_{κ}∥_{L^{p}(μ)}^{p})^{\frac{1}{p}} · (∑_{ν} ∥x_{ν}∥_{ℓ^{∞}(k)}^{2})^{\frac{1}{2}}

≤ k^{1/p+1/2}∥I∥_{p} · (∑_{ν} ∥x_{ν}∥_{ℓ^{∞}(k)}^{2})^{\frac{1}{2}} .

4.1.13 Choose p = 0.5, p_{1} = .253, p_{2} = .559, q = 0.8, β = 1/4, and δ = α/8 and evaluate on the computer T_{p,q}(E) ≤ 4.327 …, B_{[β],q}^{(A.8.13)} ≤ 17.76 … (cf. inequality (A.8.16)), and get

\frac{B_{[β],q} · T_{p,q}(E) · ∥I ∥_{[δ]}}{(αβ − δ)^{1/p}} ≤ \frac{76.87 . . . · ∥I ∥_{[α/8]}}{(α/8)^{2}} ≤ \frac{5000 ∥I ∥_{[α/8]}}{α^{2}} .

Page 208.. Exercise A.8.17 gives, with f = g′ and r = p/(q − p),

∥g′g∥_{[α;ℙ]} ≤ ∥g′∥_{L^{p/(q−p)}(ℙ/g)} · (\frac{2}{α})^{q/p} · ∥g∥_{[α/2;ℙ]}^{q/(q−p)}

≤ E_{p,q}^{(4.1.6)} · (\frac{2}{α})^{(q−p)/p} · (E_{[α/2], p}^{(4.1.9)}[ Z _{[.]}])^{q/p} .

Thus  E_{[α],q}^{(4.1.9)} ≤ E_{p,q} · (2/α)^{(q−p)/p} · (E_{[α/2],p}^{(4.1.9)}[ Z _{[.]}])^{q/p}

for any p ∈ (0, 1), q ≥ 1, and α ∈ (0, 1). 4.1.14 (i) Apply theorem 4.1.7. (ii) Reduce to (i) by proposition 4.1.12. 4.2.1 K[Z] ⊂ K[Z′] and S_{∞}[Z] ∈ K[Z]. 4.2.9 Let μ(dt) = f_{t} dy_{t} and ν(dt) = dz_{t}. Clearly μ({0}) ≤ ν({0}). Consider next an interval of the form [s, t). Given ϵ > 0 set t_{0} = s and t_{n+1} = inf{t > t_{n} : |f_{t} − f_{t_{n}}| ≥ ϵ}. Since f does not oscillate at any point, sup_{n} t_{n} > t. There will be an N such that t_{N} < t ≤ t_{N+1} . Redefine t_{N+1} = t and consider the partition {t_{0} < t_{1} … < t_{N+1} = t} of (s, t] into half-open intervals [t_{n}, t_{n+1}). Since f does not oscillate by more than ϵ on any of these intervals, we have

μ([s, t)) = ∑_{n} μ([t_{n}, t_{n+1}))

≤ ∑_{n} sup_{t_{n}≤t<t_{n+1}} f(t) · (y_{t_{n+1}} − y_{t_{n}})

≤ ϵ(y_{t} − y_{s}) + ∑_{n} inf_{t_{n}≤t<t_{n+1}} f(t) · (y_{t_{n+1}} − y_{t_{n}})

≤ ϵ(y_{t} − y_{s}) + ∑_{n}(z_{t_{n+1}} − z_{t_{n}})

= ϵ(y_{t} − y_{s}) + ν([s, t)) .

Since this is true for all ϵ > 0, we have μ ≤ ν on all intervals of the form [s, t) and their finite unions, then on the sequential closure of these, which is the Borel σ-algebra. 4.2.11 There are arbitrarily large stopping times T such that M^{T} is a global L^{1}-integrator and M_{.−}^{T} is bounded (corollary 2.5.29). Let N be a bounded martingale and X ∈ E_{1} . The first two terms in

(X∗M)_{T} ·N_{T} = ∫_{0}^{T} (X∗M)_{.−} dN + ∫_{0}^{T} X·N_{.−} dM + [(X∗M), N]_{T}

have expectation zero. Thus

𝔼[(X∗M)_{T} · N_{T} ] = 𝔼[ [(X∗M), N]_{T} ] = 𝔼[(X∗[M, N])_{T} ] ≤ 𝔼[ [M, N] _{∞} ]

by corollary 4.2.8: ≤ 2\sqrt{2p} · ∥S_{∞}[M]∥_{L^{p}} · ∥N_{∞}∥_{L^{p′}} .

Taking the supremum over N with ∥N_{∞} ∥_{L^{p′}} ≤ 1 and all X ∈ E_{1} , and letting T → ∞ yields the claim. The last statement is better done this way: for X ∈ E_{1}

𝔼[(X∗M)_{∞}^{2}] = 𝔼[[X∗M, X∗M]_{∞}] = 𝔼[(X^{2}∗[M, M])_{∞}] ≤ 𝔼[[M, M]_{∞}] .

4.2.13 Using lemma 4.2.2 on page 210, continue at inequality (4.2.7) on page 214 instead with

\begin{aligned}𝔼[|M_{T} |^{⋆q}] & ≤ \frac{eq^{2}}{2} · (𝔼[(M_{T}^{⋆(q−2)})^{q′/(2−q′)}]^{(2−q′)/q′}) · ∥M∥_{K^{q}}^{2} \\ & = \frac{eq^{2}}{2} · (𝔼[M_{T}^{⋆q}])^{(q−2)/q} · ∥M∥_{K^{q}}^{2} , \end{aligned}

divide, and take the square root. The first inequality is corollary 4.2.4. 4.2.14 In view of exercise 2.5.17 only the case 1 < q < 2 is new. We continue the notations from its answer on page 470. The crucial inequality ∥S_{n} ∥_{L^{2}} ≤ σ\sqrt{n} can be replaced by

∥S_{n}∥_{L^{q}} ≤ C_{q}^{(4.2.4)}∥[S, S]_{n}^{1/2}∥_{L^{q}} = C_{q}∥(∑_{ν=1}^{n} F_{ν}^{2})^{1/2}∥_{L^{q}}

≤ C_{q}∥(∑_{ν=1}^{n} |F_{ν}|^{q})^{1/q}∥_{L^{q}} ≤ C_{q}σ_{q}n^{1/q} .

A similar argument gives

∥|Z̃_{n}|^{⋆}∥_{L^{q}} ≤ C_{q}∥(∑_{ν=1}^{n} \frac{F_{ν}^{2}}{ν^{2}} )^{1/2}∥_{L^{q}} ≤ C_{q}∥(∑_{ν=1}^{n} \frac{|F_{ν}|^{q}}{ν^{q}} )^{1/q}∥_{L^{q}}

≤ C_{q}σ_{q}(∑ν^{−q})^{1/q}<∞ ,

showing that Z̃ is a global L^{q}-integrator and thus has almost surely a limit at infinity. The expectation of the total variation of Ẑ can be estimated by

∑_{2≤ν<∞} \frac{𝔼[|S_{ν−1}|]}{ν(ν − 1)} ≤ ∑_{2≤ν<∞} \frac{∥S_{ν−1} ∥_{L^{q}}}{ν(ν − 1)} ≤ C_{q}σ_{q} ∑_{2≤ν<∞} \frac{(ν−1)^{1/q}}{ν(ν − 1)} < ∞ .

4.2.17 Roger Sewell, <rfs@cambridgeconsultants.com>, noticed that these inequalities hold for arbitrarily L^{0}-integrators Z rather than only for martingales M, as the original formulation of the exercise would suggest. The left-hand inequality becomes obvious upon the choice T = 0 in the definition of K[Z], given our convention that [Z, Z]_{0−} = 0. For the right-hand inequality replace the estimate in the proof of inequality (4.2.5) for S =^{def} S[Z] and 2 ≤ q < ∞ by

𝔼[S_{∞}^{q}] ≤ \frac{q}{2} · 𝔼[∫_{0}^{∞} S^{q−2} d(S^{2})]

≤ \frac{q}{2} · inf \{𝔼[S^{q−2} · g^{2}] : g ∈ K^{q}[Z]\}

≤ \frac{q}{2} · (𝔼[S_{∞}^{q}])^{q/(q−2)} · inf \{(𝔼[g^{q}])^{2/q} : g ∈ K[Z]\} .

Thus ∥S_{∞}[Z]∥_{L^{q}} ≤ \sqrt{q/2} · ∥Z∥_{K^{q}} .

4.2.19 As r < p, M is an L^{r}-integrator. Since X is M-measurable (page 118) it suffices to show that ∥X ∥_{M−r}^{∗} is finite (theorem 3.4.10). Now

∥X∥_{M−r}^{∗} ≤ C_{r}^{(4.2.8)} · ∥(∫ X^{2} d[M, M])^{1/2}∥_{L^{r}}
≤ C_{r} · ∥X_{∞}^{⋆} · S_{∞}[M]∥_{L^{r}}
≤ C_{r} · ∥X_{∞}^{⋆}∥_{L^{q}} · ∥S_{∞}[M]∥_{L^{p}} < ∞ .

4.2.21 By exercise 1.3.6 on page 26, T^{c} < ∞ so that T^{c} ∧ n ↑ T^{c}. Taking the expectation in

W_{T^{c}∧n}^{2} = 2 ∫_{0}^{T^{c}∧n} W dW + T^{c} ∧ n

yields

𝔼[W_{T^{c}∧n}^{2}] = 𝔼[T^{c} ∧ n] ,

and Fatou’s lemma A.8.7 gives

𝔼[c^{2}] = 𝔼[lim inf W_{T^{c}∧n}^{2}] ≤ lim inf 𝔼[T^{c} ∧ n] = 𝔼[T^{c}] ≤ c^{2} .

T^{c+} is handled the same way. 4.2.22 Set V =^{def} ∑_{i}[M^{i}, M^{i}]. In view of the inequalities of Kunita– Watanabe (theorem 3.8.9), d[M^{i}, M^{j}] is absolutely continuous with respect to d[M^{i}, M^{i}] for 1 ≤ i, j ≤ n and then with respect to dV . Therefore there exists a symmetric positive semidefinite matrix G^{ij} of wellmeasurable processes, the Radon–Nikodym derivatives d[M^{i}, M^{j}]/dV , so that d[M^{i}, M^{j}] = G^{ij} dV . It has trace 1. The usual diagonalization process for a symmetric matrix furnishes further an orthonormal matrix U_{ν}^{i} and scalars λ_{1} ≥ λ_{2} ≥ … ≥ λ_{n} ≥ 0 so that G^{ij} = ∑_{ν} U_{ν}^{i}U_{ν}^{j}λ_{ν}. The similarity-invariance of the trace gives ∑ λ_{ν} = 1. Repeated applications of lemma A.2.21 on page 378 during the diagonalization process show that the U_{ν}^{i} and the λ_{ν} can be chosen to depend Borel measurably on G^{ij}, so that they are again well-measurable processes. Next observe that for X = (X_{i}) ∈ 𝔏^{1}[M−p]

X∗M _{I^{p}} ∼ ∥S_{∞}[X∗M]∥_{L^{p}}

= ∥(∫ ∑_{i,j=1}^{n} X_{i}X_{j} d[M^{i}, M^{j}])^{1/2}∥_{L^{p}}

= ∥(∫ ∑_{i,j} X_{i}X_{j} G^{ij} dV )^{1/2}∥_{L^{p}}

= ∥(∫ ^{∗} ∑_{ν}⟨X|U_{ν}⟩^{2}λ_{ν} dV )^{1/2}∥_{L^{p}}^{∗}

= ∥⟨X|U.⟩∥^{∗} ,

where  ∥F ∥^{∗} =^{def} ∥(∫ ^{∗} ∑_{ν=1}^{n} |F_{ν}|^{2}λ_{ν} dV )^{1/2}∥_{L^{p}}^{∗}

on functions F = (F_{ν}) that live on B̌ =^{def} {1, …, n} × B. After these preliminaries, let (^{1}X,^{2}X,^{3}X …) be a sequence in 𝔏^{1}[M−p] such that ^{n}X∗M converges in H_{0}^{p} to some martingale L. Then ^{n}F (ν, ϖ) =^{def} ⟨^{n}X(ϖ)|(U_{ν})(ϖ)⟩ defines a Cauchy sequence (^{n}F) for the mean ∥ ∥^{∗}, and replacing (^{n}X) by a subsequence we may in view of theorem 3.2.22 assume that ⟨^{n}X|U.⟩ converges both ∥ ∥^{∗}-a.e. and in ∥ ∥^{∗}-mean to some function F on B̌. Since

∑_{ν} U_{ν}^{i}U_{ν}^{j} = δ^{ij}, ^{n}X_{i} = ∑_{ν}⟨^{n}X|U_{ν}⟩U_{ν}^{i} → X_{i} =^{def} ∑_{ν} F_{ν}U_{ν}^{i} both ∥ ∥^{∗}-a.e.

and in ∥ ∥^{∗}-mean, for i = 1, …, n. For ease of thinking and without loss of generality (see page 116) we assume the ^{n}X_{i} and X_{i} are predictable. Now in case a) G^{ij} is diagonal, U_{ν}^{i} = δ_{ν}^{i}, X ↦ ∥⟨X|U.⟩∥^{∗} is solid and thus is a mean; in fact, it is equivalent to the Daniell mean \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{M−p}^{∗} on previsibles (proposition 3.6.1 and exercise 3.6.16). Then X ∈ 𝔏^{1}[M−p], and L = X∗M. In case b) the brackets [M^{i}, M^{j}] are previsible, so are the G^{ij}, λ_{ν}, and the U_{ν}^{i}; we set N_{ν} =^{def} U_{ν}∗M, obtaining martingales satisfying a) and having {N^{1}, …, N^{n}}^{∥} = A^{∥}. Now an L ∈ A^{∥} can be written L = ∑_{ν} X^{ν}∗N^{ν} = (∑_{ν} X^{ν}U_{ν}^{i})∗M^{i}. In case c) we replace the [M^{i}, M^{j}] by the previsible brackets ⟨M^{i}, M^{j}⟩ (see page 228), obtain previsible G^{ij}, λ_{ν}, and U_{ν}^{i}, and continue as in b). 4.2.23 We prove the very last statement: (A^{⊥ ⊥})^{⊥ ⊥} = A^{∥}. The inclusion A^{∥} ⊆ (A^{⊥ ⊥})^{⊥ ⊥} is obvious. For the converse suppose N ∈ H_{0}^{p} lies outside A^{∥}. The general theorem of Hahn–Banach (A.2.25 (i)) provides a martingale N^{∗} in the dual H_{0}^{p∗} with ⟨M|N^{∗}⟩ ≤ 1 for all M ∈ A^{∥} and ⟨N|N^{∗}⟩ > 1. Here ⟨ | ⟩ denotes the duality. Since A^{∥} is a subspace, ⟨M|N^{∗}⟩ = 0 for all M ∈ A^{∥} so that N^{∗} ∈ (A^{∥})^{⊥ ⊥} ⊆ A^{⊥ ⊥}. Since ⟨N|N^{∗}⟩ > 1, N also lies outside (A^{⊥ ⊥})^{⊥ ⊥}. 4.3.5 A process Z such that {Z_{T} : T ∈ 𝔗[F.] , T < ∞} is uniformly integrable is called a process of class (D) in the literature ([73, page 101]); it is called a process of class (DL) if the stopped process Z^{u} is of class (D) for all instants u < ∞ (ibidem).

Suppose then that Z is a positive supermartingale that is continuous in probability. By lemma 2.5.27 (ii) on page 80, Z ∧ c is a global L^{2}-integrator with a càdlàg modification, for any c > 0. We may thus assume that Z itself has càdlàg paths and a limit Z_{∞} ∈ \overline{ℝ} at ∞.

The necessity of (D) for the conclusion is obvious: The global integrator Ẑ is of class (D) since |Ẑ_{T} | ≤ Ẑ_{∞}^{⋆} ∈ L^{1} for all T ∈ 𝔗[F.] (see theorem 2.3.6 on page 63); and Z̃_{t} = 𝔼[Z̃_{∞}|F_{t}] is of class (D) by example 2.5.2 on page 72. Thus their sum Z is of class (D) as well.

To show the sufficiency, assume that Z is of class (D). Then Z_{∞} = lim_{t→∞} Z_{t} also in L^{1}-mean.

An aside: The martingale M_{t} =^{def} 𝔼[Z_{∞}|F_{t}] is of class (D) as well (ibidem), and Z′ =^{def} Z − M is a positive supermartingale of class (D) with 𝔼[Z′_{t}] ⟶_{t→∞} 0.

A positive supermartingale Z′ with 𝔼[Z′_{t}] ⟶_{t→∞} 0 is called a potential. Thus a positive supermartingale Z of class (D) can be written as the sum of a potential Z′ of class (D) and a uniformly integrable martingale M. This decomposition Z = Z′ + M is known as the Riesz decomposition of Z.

For any c > 0 set T_{c} =^{def} inf{t : Z_{t} ≥ c}. Then Z^{T_{c}} ∧ c is a global L^{2}-integrator (see lemma 2.5.27 (ii)) that differs by the global L^{1}-integrator (Z_{T_{c}} − c)_{+} · [[T_{c}, ∞)) from Z^{T_{c}}. The latter is therefore a global L^{1}-integrator. Now B =^{def} [sup_{c} T_{c} < ∞] is negligible: if it were not, then the random variables Z_{T_{c}} ≥ c · B, c < ∞, would not be uniformly integrable. This means that T_{c} ⟶_{c→∞} ∞ and implies that Z is a local L^{1}-integrator and has a Doob–Meyer decomposition Z = Ẑ + Z̃ (with Ẑ_{0} = Z_{0} and Z̃_{0} = 0 by our convention). Since μ_{Z} ≤ 0, Ẑ is a decreasing process; since 𝔼[Ẑ_{t} − Ẑ_{0}] = 𝔼[Z_{t} − Z_{0}] ≥ 𝔼[Z_{∞} − Z_{0}], we have 𝔼[Ẑ_{t}] ≥ 𝔼[Z_{∞}] for all t, so Ẑ_{∞} =^{def} lim_{t→∞} Ẑ_{t} exists almost surely and in L^{1}-mean. Therefore Ẑ is a global L^{1}-integrator of size Ẑ _{I^{1}} ≤ 𝔼[Z_{0}] + 𝔼[Z_{0} − Z_{∞}] and is of class (D). Then Z̃ = Z − Ẑ is a martingale of class (D) as well. The sufficiency of the condition (D) is established.

An aside: Assume that Z is a potential and set M =^{def} Z̃ + Z_{0} and A =^{def} Z_{0} − Ẑ. Then M is a uniformly integrable martingale with M_{0} = Z_{0} and A is an increasing predictable process with A_{0} = 0 and 𝔼[A_{∞}] < ∞; and Z = M −A. Since Z_{∞} = 0, we have M_{∞} = A_{∞} and M_{t} = 𝔼[A_{∞}|F_{t}]. That is to say, Z_{t} = 𝔼[A_{∞}|F_{t}] − A_{t} is determined entirely by the predictable increasing process A; it is called the potential generated by A.

By applying the result above to the stopped processes Z^{u}, u < ∞, we get the following: For a positive supermartingale Z to have a Doob–Meyer decomposition Z = Ẑ + Z̃ with Ẑ of integrable finite variation ( Z _{t} ∈ L^{1} ∀ t < ∞) and Z̃ a martingale (rather than merely a local one) it is necessary and sufficient that Z be of class (DL).

By applying it to the stopped processes Z^{U}, U a finite stopping time, we get the following: For a positive supermartingale Z to have a Doob–Meyer decomposition it is necessary and sufficient that Z be locally of class (D), i.e., Z^{U} to be of class (D) for arbitrarily large stopping times U. 4.3.6 Let (S^{(n)}) be a sequence of stopping times that announces T. Then

𝔼[Z̃_{T} |F_{S^{(n)}}] = Z̃_{S^{(n)}} ⟶_{n→∞} Z̃_{.−}_{T} and thus 𝔼[Z̃_{T} − Z̃_{.−}_{T} |F_{T−}] = 0. Since

Ẑ_{T} ∈ F_{T−}, Ẑ_{T} − Ẑ_{.−}_{T} =𝔼[Ẑ_{T} − Ẑ_{.−}_{T} |F_{T−}]=𝔼[Z_{T} − Z_{T−}|F_{T−}]. Thanks to Roger Sewell, <rfs@cambridgeconsultants.com> (12/21/2006), who noticed that the result extends to more general circumstances: for the various conditional expectations above to make sense it is not necessary that Z be a global L^{1}-integrator, That T reduce it to one will do. This happens, for instance, if Z is a plain L^{1}-integrator and T is bounded. In fact, the equality ΔẐ_{T} = 𝔼 [ΔZ_{T} |F_{T−}] holds whenever Z is a local L^{1}-integrator and ΔZ_{T} is integrable. To see this, let U_{n} be a sequence of stopping times that reduce Z to a global L^{1}-integrator and increase to ∞. Let B =^{def} A ∩ [t < T], A ∈ F_{t}, be a typical generator of F_{T−} (as ΔZ_{0} = ΔẐ_{0} we need not worry about t = 0). Then [T < U_{n}] = ⋃_{q∈ℚ}[T ∧ U_{n} ≤ q] ∩ [q < U_{n}] belongs to F_{T ∧U_{n}−} and is contained in [T ∧ U_{n} = T], B_{n} =^{def} A ∩ [t < T ∧ U_{n}] ∩ [T < U_{n}] ∈ F_{T ∧U_{n}−} is contained in [T = T ∧ U_{n}] as well, and therefore

𝔼[ΔZ_{T} · B_{n}] = 𝔼[ΔZ_{T ∧U_{n}}^{U_{n}} · B_{n} ] = 𝔼[ΔẐ_{T ∧U_{n}}^{U_{n}} · B_{n} ] = 𝔼[ΔẐ_{T} · B_{n} ] .

As n → ∞, the Dominated Convergence Theorem gives 𝔼[ΔZ_{T} · B] = 𝔼[ΔẐ_{T} · B]. As ΔẐ_{T} ∈ F_{T−}, we get ΔẐ_{T} = 𝔼 [ΔZ_{T} |F_{T−}].

For the second claim reduce to global L^{1}-integrators; at the predictable time T^{ϵ} = inf{t : ΔẐ_{t} ≥ ϵ} the jump is zero. Thus T^{ϵ} = ∞.

To prove the inequality, let M be a martingale with ∥M_{∞} ∥_{(q/2)′} ≤ 1. There is a countable family {T_{n}} of stopping times, all of them necessarily predictable, at which the jumps of Ẑ occur (theorem 2.4.4). Then

𝔼[[Ẑ, Ẑ]_{∞} · M_{∞}] = 𝔼[∑_{n} |𝔼[ΔZ_{T_{n}}|F_{T_{n}−}]|^{2} · M_{∞}]
by inequality (A.3.10):  ≤ 𝔼[∑_{n} 𝔼[(ΔZ_{T_{n}})^{2}|F_{T_{n}−}] · M_{T_{n}−}]

= 𝔼[∑_{n}(ΔZ_{T_{n}})^{2} · M_{T_{n}−}] ≤ 𝔼[^{j}[Z, Z]_{∞} · M_{∞}^{⋆}]

by theorem 2.5.19:  ≤ ∥^{j}[Z, Z]_{∞}∥_{q/2} · (q/2) = (q/2) ∥^{j}S_{∞}[Z]∥_{q}^{2} .

Taking the supremum over M and then the square root gives the claim. 4.3.8 Reduce to the case that Z is a global L^{1}-integrator and let M be a bound for the jumps of Z, let Z = Ẑ + Z̃ be the Doob–Meyer decomposition of Z and S a stopping time, necessarily predictable (exercise 3.5.19), at which Ẑ jumps. In view of exercise 4.3.6 we have |ΔẐ_{S}| ≤ M, and therefore |ΔZ̃_{S}| ≤ 2M. Thus |ΔZ̃| ≤ 2M. Let K > 0 and U = inf{t : Ẑ _{t} ∨ S_{t}[Z̃] ≥ K}. U can be made arbitrarily large by the choice of K. Then Ẑ^{U} is a global L^{q}-integrator for all q since ∥ Ẑ^{U} _{∞} ∥_{L^{∞}} ≤ K + M, and Z̃^{U} is a global L^{q}-integrator for all q since ∥S_{∞}[Z̃^{U}]∥_{L^{∞}} ≤ K +2M (exercise 4.2.18). 4.3.9 We may without loss of generality assume that Z is a global L^{1}-integrator. Let U = inf{t : |ΔZ_{t}| ≥ M}. This stopping time can be made arbitrarily large by the choice of M > 0. Let Z^{U−} be the process Z stopped just before U:

Z^{U−} = Z · [[0, U)) + Z_{U−} · [[U, ∞)) = (Z − ΔZ_{U})^{U} .

Since Z^{U−} differs by the L^{1}-bounded finite variation process ΔZ_{U} · [[U, ∞)) from the L^{1}-integrator Z^{U}, it is a global L^{1}-integrator itself, with jumps uniformly bounded by M. Now apply exercise 4.3.8. 4.3.12 (i)⇒(ii) Let N be a previsible set with μ_{V′}(N) = 0. By Fubini’s theorem A.3.18 we have ∫_{N} dV′_{t}(ω) = 0 and then ∫_{N} dV_{t}(ω) = 0 for almost all ω ∈ Ω and, taking the expectation, μ_{V} (N) = 0.

(ii)⇒(iii) is immediate from the theorem of Radon–Nikodym.

It is left to prove the last statement under the assumption (iii); it will entrain the implication (iii)⇒(i). Let \overline{V} = G∗V′. This is a previsible (exercise 3.7.19) positive increasing process. Both V and \overline{V} are Doléans–Dade processes for the measure μ_{V} , so they are indistinguishable. 4.3.13 Let D be a Radon–Nikodym derivative of μ with respect to the variation measure μ . This is a previsible process of absolute value one. Since μ = D · μ and μ ≥ μ we have V ≥ V = D∗V . On the other hand, d(D∗V) is a positive measure on the line majorizing |dV |, so D∗V ≥ V . To prove the remaining inequality V _{I^{p}} ≥ ∥ V _{∞} ∥_{L^{p}}, note that V _{I^{p}} ≥ ∥∫ D dV ∥_{L^{p}} = ∥ V _{∞} ∥_{L^{p}}.

4.3.14 There is an increasing sequence of stopping times T_{n} so that M^{(n)} =^{def} M^{T_{n+1}} − M^{T_{n}} is the sum of a finite variation process V^{(n)} and a global L^{2}-integrator Z^{(n)} (corollary 2.5.29). Let Z^{(n)} = Ẑ^{(n)} + Z̃^{(n)} be the Doob–Meyer decomposition and write

M = ∑_{n}(V^{(n)} + Ẑ^{(n)}) + ∑_{n} Z̃^{(n)} .

4.3.15 By corollary A.5.13 on page 438 the maximal process X^{⋆} is progressively measurable and therefore adapted. Since it is increasing it has a càdlàg version X_{.+}^{⋆}, which is a global L^{p}-integrator iff ∥X_{∞}^{⋆} ∥_{L^{p}} is finite. (Thanks to Roger Sewell, who added this remark and further improvements to my previous slapdash version of this answer.) First the case r ≥ 1, which implies that q > 1. Then Z has a Doob–Meyer decomposition Z = Ẑ + Z̃,

∫ X dZ = ∫ X dẐ + ∫ X dZ̃ ≤ X_{∞}^{⋆} · Ẑ _{∞} + ∫ X dZ̃ ,
and ∥∫ X dZ∥_{L^{r}} ≤ ∥X_{∞}^{⋆} · Ẑ _{∞}∥_{L^{r}} + ∥∫ X dZ̃∥_{L^{r}}
by A.8.4:  ≤ ∥X_{∞}^{⋆}∥_{L^{p}} · ∥ Ẑ _{∞}∥_{L^{q}} + C_{r}^{(4.2.4)} · ∥S_{∞}[X∗Z̃]∥_{L^{r}}

= ∥X_{∞}^{⋆}∥_{L^{p}} · ∥ Ẑ _{∞}∥_{L^{q}} + C_{r} · ∥(∫_{0}^{∞} X^{2} d[Z̃, Z̃])^{1/2}∥_{L^{r}}

≤ ∥X_{∞}^{⋆}∥_{L^{p}} · Ẑ _{I^{q}} + C_{r} · ∥X_{∞}^{⋆} · S_{∞}[Z̃]∥_{L^{r}}
≤ ∥X_{∞}^{⋆}∥_{L^{p}} · Ẑ _{I^{q}} + C_{r} · ∥X_{∞}^{⋆}∥_{L^{p}} · ∥S_{∞}[Z̃]∥_{L^{q}}
≤ ∥X_{∞}^{⋆}∥_{L^{p}} · Ĉ_{q}^{(4.3.1)} Z _{I^{q}}
+ C_{r} · ∥X_{∞}^{⋆}∥_{L^{p}} · K_{q}^{(3.8.6)} Z̃ _{I^{q}}
≤ C_{q,r} · ∥X_{∞}^{⋆}∥_{L^{p}} · Z _{I^{q}}
+ C_{q,r} · ∥X_{∞}^{⋆}∥_{L^{p}} · C̃_{q}^{(4.3.1)} Z _{I^{q}}
= C_{q,r} · ∥X_{∞}^{⋆}∥_{L^{p}} · Z _{I^{q}}  (∗)

— the constants C_{r} and C_{q,r} above are understood to vary from one occurrence to the next. Now if r < 1 pick a q′ ≥ q/r > 1 and let ℙ′ be a probability equivalent to ℙ under which both global L^{q}-integrators X_{.+}^{⋆p/q} and Z are global L^{q′}-integrators. By theorem 4.1.2 on page 191, such ℙ′ exists if ∥X_{∞}^{⋆} ∥_{L^{p}} < ∞, the only case where there is anything to prove.

Now  \frac{q}{rq′} = \frac{q}{pq′} + \frac{1}{q′} , i. e., \frac{1}{rq′/q} = \frac{1}{pq′/q} + \frac{1}{q′} ,

and as 1 ≤ rq′/q < pq′/q we may apply the case treated above:

∥∫ X dZ∥_{L^{r}(ℙ)} ≤ (E_{q,q′}^{(4.1.7)})^{q/q′r} · ∥∫ X dZ∥_{L^{rq′/q}(ℙ′)}
≤ E_{...} · C_{q′,r}^{(∗)} · ∥X_{∞}^{⋆}∥_{L^{pq′/q}(ℙ′)} · Z _{I^{q′} (ℙ′)}
= C_{...} · (∫ (X_{∞}^{⋆p/q})^{q′} dℙ′)^{q/pq′} · D_{q,q′,2}^{(4.1.5)} Z _{I^{q}(ℙ)}
= C_{...} · X_{.+}^{⋆p/q} _{I^{q′} (ℙ′)}^{q/p} · Z _{I^{q}(ℙ)}
≤ C_{...} · D_{q,q′,2}^{(4.1.5)} X_{.+}^{⋆p/q} _{I^{q}(ℙ)}^{q/p} · Z _{I^{q}(ℙ)}
= C_{...} · ∥X_{∞}^{⋆p/q}∥_{L^{q}(ℙ)}^{q/p} · Z _{I^{q}(ℙ)}
= C_{...} · ∥X_{∞}^{⋆}∥_{L^{p}(ℙ)} · Z _{I^{q}(ℙ)} .

Applying this to Y X with Y ∈ E_{1} gives

∥∫ Y X dZ∥_{L^{r}} = ∥∫ Y d(X∗Z)∥_{L^{r}} ≤ C_{p,q} · ∥X_{∞}^{⋆}∥_{L^{p}} · Z _{I^{q}} ,
whence  X∗Z _{I^{r}} ≤ C_{p,q} · ∥X_{∞}^{⋆}∥_{L^{p}} · Z _{I^{q}} ,  0 < \frac{1}{r} = \frac{1}{p} + \frac{1}{q} < ∞ .

4.3.17 Copy the proofs of the corresponding statements for the square bracket. 4.3.18 Use exercise 3.8.24 on page 157 (ii). 4.3.20 If p ≤ 2, argue that, for X ∈ E_{1} , ∥X∗M^{t}∥_{L^{p}} ≤ ∥(X∗M^{t})^{⋆}∥_{L^{p}} ≤

C_{p}^{(4.3.6)} · ∥s_{∞}[(X ∗ M^{t})]∥_{L^{p}} =C_{p} · ∥(∫_{0}^{t} X^{2} d⟨M, M⟩)^{1/2}∥_{L^{p}} ≤C_{p} · ∥s_{t}[M]∥_{L^{p}} =

C_{p} · ∥s_{∞}[M^{t}]∥_{L^{p}} implies M^{t} _{I^{p}}≤C_{p} · ∥s_{t}[M]∥_{L^{p}}. If p ≥ 2, use the fact that s[M] = S[M] as [M, M] is previsible, and employ inequality (4.2.4) on page 213 instead. 4.3.21 Taking the expectation in

∫ X dZ ≤ ∫ |X| d Ẑ + ∫ X dZ̃ ≤ X_{∞}^{⋆} · Ẑ _{∞} + ∫ X dZ̃

gives ∥Z ∥_{p}^{∧} ≤ ∥ Ẑ _{∞} ∥_{L^{p}}. The measurability of X_{∞}^{⋆} is not required for this argument. The remaining inequality ∥ Ẑ _{∞} ∥_{L^{p}} ≤ p·∥Z ∥_{p}^{∧} is nontrivial only if ∥Z ∥_{p}^{∧} < ∞. In this case |Ẑ_{0}| = ∫ sgn(Ẑ_{0})·[[0]] dZ is p-integrable. There are then arbitrarily large stopping times S such that Ẑ^{S} = Ẑ ^{S} is bounded (see corollary 3.5.16). Let D = D^{−1} be a previsible derivative of the Doléans– Dade measure ν_{Z} with respect to its variation and set X =^{def} p·D· Ẑ^{S} ^{p−1} . This previsible process has its càdlàg maximal process in L^{p′}:

∥X_{∞}^{⋆} ∥_{L^{p′}} = ∥X_{∞} ∥_{L^{p′}} = p·(𝔼[ Ẑ _{S}^{(p−1) \frac{p}{p−1}} ])^{\frac{p−1}{p}} = p · ∥ Ẑ _{S} ∥_{L^{p}}^{p−1} .

Now d Ẑ ^{p} ≤ p Ẑ ^{p−1}d Ẑ = pD Ẑ ^{p−1}dẐ = X dẐ on [[0, S]] and consequently

∥ Ẑ _{S}∥_{L^{p}}^{p} ≤ 𝔼[∫ X dẐ^{S}] ≤ p · ∥ Ẑ _{S}∥_{L^{p}}^{p−1} · ∥Z∥_{p}^{∧} .

Division by the middle factor yields

∥ Ẑ _{S}∥_{L^{p}} ≤ p · ∥Z∥_{p}^{∧} .

Now take S ↑ ∞ to arrive at the claim. Thanks to Roger Sewell, <rfs@cambridgeconsultants.com> for pointing out various typos in the first version (01/16/2007). 4.3.22 Take the supremum over g ∈ L_{1}^{p′} over

𝔼[Î_{∞} · g] = 𝔼[∫_{0}^{∞} M_{.−}^{g} dÎ] = 𝔼[∫_{0}^{∞} M_{.−}^{g} dI]
≤ 𝔼[M_{∞}^{g⋆} · I_{∞}] ≤ ∥M_{∞}^{g⋆}∥_{L^{p′}} · ∥I_{∞}∥_{L^{p}}
by theorem 2.5.19:  ≤ p · ∥I_{∞}∥_{L^{p}} = p · I _{I^{p}} .

4.3.27 Applying the expectation in equation (3.10.6) on page 182 gives (using the Einstein convention)

𝔼[Φ(Z_{t})] = 𝔼[Φ(Z_{0})]

+ 𝔼[∫_{0+}^{t} Φ_{;η}(Z_{.−}) dẐ^{η}]

+ \frac{1}{2} 𝔼[∫_{0+}^{t} Φ_{;ηθ}(Z_{.−}) d^{c}[Z^{η}, Z^{θ}]]

+ 𝔼[∫_{0}^{t}(Φ(Z_{s−} + y)−Φ(Z_{s−})−Φ_{;η}(Z_{s−}) · y^{η}) ĵ_{Z}(dy, ds)] .

To see that this clumsy looking formula might have some interest, consider the case that Z is a Lévy process. Then Ẑ_{t}^{η} = A^{η} × t, ^{c}[Z^{η}, Z^{θ}]_{t} = B^{ηθ} × t, and ĵ_{Z}(dy, ds) = ν(dy)×ds, with constant vector A^{η}, constant matrix B^{ηθ}, and constant measure ν(dy) on ℝ^{d} (lemma 4.6.7 and lemma 4.6.8 on page 258). Then we get the more informative formula

𝔼[Φ(Z_{t})] = 𝔼[Φ(Z_{0})]

+ ∫_{0}^{t} 𝔼[A^{η}Φ_{;η}(Z_{s−})] ds

+ \frac{1}{2} ∫_{0}^{t} 𝔼[B^{ηθ}Φ_{;ηθ}(Z_{s−})] ds

+ ∫_{0}^{t} ∫_{ℝ^{d}} 𝔼[Φ(Z_{s−} + y)−Φ(Z_{s−})−Φ_{;η}(Z_{s−}) · y^{η}] ν(dy) ds ,

or, as Z_{s−} = Z_{s} almost surely at all instants s,

\frac{d𝔼[Φ(Z_{t})]}{dt} = A^{η} 𝔼[Φ_{;η}(Z_{t})] + (B^{ηθ}/2) 𝔼[Φ_{;ηθ}(Z_{t})]
+ ∫_{ℝ^{d}} ν(dy) 𝔼[Φ(Z_{t} + y)−Φ(Z_{t})−Φ_{;η}(Z_{t}) · y^{η}] .

Let us assume that Z starts with Z_{0} = 0, and consider the function (t, x) ↦

u(t, x) =^{def} 𝔼[Φ(x + Z_{t})]. Then, applying the previous equality to the function

z ↦ Φ(x + z), we see that u solves the initial value problem

u(0, x) = Φ(x) ,
\frac{du(t, x)}{dt} = A^{η} u_{;η}(t, x) + (B^{ηθ}/2) u_{;ηθ}(t, x)
+ ∫_{ℝ^{d}}(u(t, x + y)−u(t, x)−u_{;η}(t, x) · y^{η}) ν(dy) .

(Thanks to Roger Sewell, <rfs@cambridgeconsultants.com>, for spotting a typo here in the first answer given.) Since to a given suitable triple (A, B, ν) a process with characteristic triple (tA, tB/2, ν(dy)dt) exists (theorem 4.6.17 on page 267), we know now that this initial value problem has a solution for any Φ ∈ C_{b}^{2} . Or, in other words, the clumsy formula above leads to a description of the generator of a Lévy process. 4.4.1 We have shown that for every n ∈ ℕ there are a stopping time T_{n} with ℙ[T_{n} < n] < 2^{−n}, a process ^{n}V of finite variation, and a L^{2}-bounded martingale ^{n}M with |Δ^{n}M| ≤ 1, both stopping at time T_{n}, such that Z^{T_{n}} = ^{n}V + ^{n}M. Replacing if necessary T_{n} by T′_{n} = inf{T_{ν} : ν ≥ n} and ^{n}V,^{n}M by the stopped processes ^{n}V^{T′_{n}},^{n}M^{T′_{n}}, we may assume that the T_{n} increase to ∞. It looks as though the right-continuity of the filtration is used here (see exercise 1.3.30); it is left to the reader to show that its regularity suffices. The decomposition

\begin{aligned}Z^{T_{n+1}} − Z^{T_{n}} & = (^{n+1}V −^{n+1}V^{T_{n}}) + (^{n+1}M −^{n+1}M^{T_{n}}) \\ & =^{def} ^{n}′V +^{n}′M \end{aligned}

has the property that its finite variation and martingale parts stop at T_{n+1} and vanish on [[0, T_{n}]]. We get the required decomposition Z = V + M by setting

V = ^{1}V + ∑_{n=1}^{∞}^{n}′V and M = ^{1}M + ∑_{n=1}^{∞}^{n}′M ,

sums which at every point ϖ ∈ B have only finitely many non-zero terms.

4.4.4 By proposition 3.7.33 F is V−0-integrable and by exercise 3.6.16 and definition (4.2.9) F is M−2-integrable; now apply exercise 3.6.14. 4.4.6 There is a countable ℚ-algebra C ⊂ C_{00}(H) whose sequential closure is B^{•}(H). For every ϕ ∈ C let P^{ϕ} be the sparse predictable support of ϕ∗ζ and set P = ⋃_{ϕ∈C} P^{ϕ}. Let S be a predictable stopping time. Then so is the time S′ whose graph is [[S]] \ P (theorem 3.5.13). The Ȟ ∈ P̌ with the property that Δ(Ȟ∗ζ)_{S′} is nearly zero is sequentially closed and contains ϕX for all ϕ ∈ C and all X ∈ E (equation (3.10.2)). Then it contains P̌. 4.5.3 Let us write ∥X ∥_{T}^{⋄} for the right-hand side of (4.5.1). To start with assume T reduces Z to a global L^{p}-integrator. For any bounded Y ∈ P^{d} with |Y_{η}| ≤ |X_{η}| clearly ∥∫ Y dZ^{T} ∥_{L^{p}} ≤ ∥X ∥_{T}^{⋄} . Corollary 3.6.10 implies that X_{η} is Z_{η}−p-integrable for 1 ≤ η ≤ d. Setting X^{(n)} =^{def} X · [|X| ≤ n],

we have (X − X^{(n)})∗Z^{T} _{I^{p}} ≤ ∑_{η} ∥X_{η} − X_{η}^{(n)} ∥_{Z^{T}−p}^{∗} ⟶_{n→∞} 0 and by theo-

rem 2.3.6 ∥(X − X^{(n)})∗Z_{T}^{⋆} ∥ ⟶_{n→∞} 0. Since ∥|X^{(n)}∗Z|_{T}^{⋆} ∥_{L^{p}} ≤ ∥X ∥_{T}^{⋄} , we

have ∥|X∗Z|_{T}^{⋆} ∥_{L^{p}} ≤ ∥X ∥_{T}^{⋄} in the limit. In general let T_{n} be stopping times reducing Z to global L^{p}-integrators. Then the previous argument gives ∥|X∗Z|_{T ∧T_{n}}^{⋆} ∥_{L^{p}} ≤ ∥X ∥_{T}^{⋄} , which produces (4.5.1) in the limit as n → ∞. 4.5.9 From (1 + c)^{x} − 1 ≥ x ln(1 + c) we get, with x = 1/p,

((1 + (1/p′)^{p})^{1/p} − 1) ≥ \frac{1}{p} ln (1 + (1 − 1/p)^{p}) .

Since p ↦ (1 − 1/p)^{p} increases this exceeds p^{−1} ln (1 + (1/2)^{2}) ≥ 1/(5p). 4.5.11 Since d[Z, Z] ≤ dΛ, Itô’s formula gives

|X∗f(Z)|_{T}^{⋆} ≤ |Xf′(Z)∗Z|_{T}^{⋆} + |Xf′′(Z)∗[Z, Z]|_{T}^{⋆} /2 , which implies
∥|X∗f(Z)|_{T}^{⋆} ∥_{L^{p}} ≤ C_{p}^{⋄}L max_{ρ=1^{⋄},2} ∥(∫_{0}^{T} |X|_{s}^{ρ} dΛ_{s})^{1/ρ}∥_{L^{p}} + \frac{L}{2}∥(∫_{0}^{T} |X|_{s} dΛ_{s})∥_{L^{p}} .

4.5.12 The Doléans–Dade measure of Z^{⟨1⟩} is the Doléans–Dade measure of 1X∗Z and by inequality (4.5.21) majorizes the Doléans–Dade measure of any other X′∗Z with X′ ∈ E_{1}^{d}. 4.5.13 The differential form of equation (4.5.23) exhibits the Doléans–Dade measure of Z^{⟨2⟩} as the Doléans–Dade measure of [^{2}X∗Z,^{2}X∗Z]. 4.5.14 Set ^{q}Ȟ =^{def} ⟨^{q}X_{s}|y⟩. The Doléans–Dade measure of |^{q}Ȟ|^{q}∗ȷ_{Z} coincides by equation (4.5.25) with that of Z^{⟨q⟩} and majorizes by definition the Doléans–Dade measures of all other |Ȟ|^{q}∗ȷ_{Z}, Ȟ as indicated. 4.5.20 (i) Inequality (4.5.27) is obvious with C_{0} = 1 when ℓ = 0. Since |d[Z^{η}, Z^{θ}]| ≤ dΛ , (4.5.28) follows from

∥∫_{T^{κ}}^{T^{μ}} g_{ι}·|Z^{ι} − Z^{ιT^{κ}} |^{⋆ℓ} |d[Z^{η}, Z^{θ}]|∥_{L^{p}} ≤ ∥∫_{T^{κ}}^{T^{μ}} g_{ι} · |Z^{ι} − Z^{ιT^{κ}} |_{s}^{⋆ℓ} dΛ_{s}∥_{L^{p}}
by theorem 2.4.7:  = ∥∫_{κ}^{μ} g_{ι} · |Z^{ι} − Z^{ιT^{κ}} |_{T^{λ}}^{⋆ℓ} dλ∥_{L^{p}}

by exercise A.3.29:  ≤ ∫_{κ}^{μ} ∥g_{ι} · |Z^{ι} − Z^{ιT^{κ}}|_{T^{λ}}^{⋆ℓ} ∥_{L^{p}} dλ
by inequality (4.5.27):  ≤ C_{ℓ} · ∥g∥_{L^{p}} ∫_{κ}^{μ}(λ−κ)^{ℓ/2} dλ
≤ \frac{1}{ℓ/2 + 1}C_{ℓ} (μ−κ)^{ℓ/2+1} · ∥g∥_{L^{p}} .(C.1)

For ℓ = 1, g_{η} · |Z^{η} − Z^{ηT^{κ}}|^{⋆} equals |g_{η} · ((T^{κ}, T^{μ}]]∗Z^{η}|^{⋆}, and theorem 4.5.1 gives

∥g_{η} · |Z^{η} − Z^{ηT^{κ}}|_{T^{μ}}^{⋆} ∥_{L^{p}} ≤ C_{p}^{⋄} · max_{ρ=1,2} ∥(∫_{T^{κ}}^{T^{μ}} |g|^{ρ} dΛ)^{1/ρ}∥_{L^{p}}
by theorem 2.4.7:  ≤ C_{p}^{⋄} · ∥g∥_{L^{p}} · max_{ρ=1,2}(μ−κ)^{1/ρ}
since μ−κ ≤ 1:  ≤ C_{p}^{⋄} (μ−κ)^{1/2} · ∥g∥_{L^{p}} .

Let then ℓ > 1 and assume inequality (4.5.27) has been established for indices up to ℓ. Now

(Z^{η}−Z^{ηT^{κ}})_{t}^{ℓ+1} = (ℓ+1)(((T^{κ}, T^{μ}]](Z^{η}−Z^{ηT^{κ}})^{ℓ}∗Z^{η})_{t}

+ \frac{(ℓ+1)ℓ}{2} ∫_{T^{κ}}^{t}(Z^{η}−Z^{ηT^{κ}})^{ℓ−1} d[Z^{η}, Z^{η}]

≤ (ℓ+1)|((T^{κ}, T^{μ}]](Z^{η}−Z^{ηT^{κ}})^{ℓ}∗Z^{η}|_{t}^{⋆}

+ \frac{(ℓ+1)ℓ}{2} ∫_{T^{κ}}^{t} |Z^{η}−Z^{ηT^{κ}} |^{ℓ−1} |d[Z^{η}, Z^{η}]|

implies

|Z^{η}−Z_{T^{κ}}^{η}|_{T^{μ}}^{⋆ℓ+1} ≤ (ℓ+1)|((T^{κ}, T^{μ}]](Z^{η}−Z^{ηT^{κ}})^{ℓ}∗Z^{η}|_{T^{μ}}^{⋆}
+ \frac{(ℓ+1)ℓ}{2} ∫_{T^{κ}}^{T^{μ}} |Z^{η}−Z^{ηT^{κ}} |^{ℓ−1} |d[Z^{η}, Z^{η}]| , (C.2)

the first term of which contributes not more than

(ℓ+1)C_{p}^{⋄} · max_{ρ=1,2} ∥(∫_{T^{κ}}^{T^{μ}}|g_{η}|Z^{η}−Z^{ηT^{κ}}|_{s}^{⋆ℓ}|^{ρ} dΛ_{s})^{1/ρ}∥_{L^{p}}

= (ℓ+1)C_{p}^{⋄} · max_{ρ=1,2} ∥(∫_{κ}^{μ} |g_{η}|Z^{η}−Z^{ηT^{κ}}|_{T^{λ}}^{⋆ℓ} |^{ρ} dλ)^{1/ρ}∥_{L^{p}}

by A.3.29:  ≤ (ℓ+1)C_{p}^{⋄} · max_{ρ=1,2} (∫_{κ}^{μ} ∥g_{η}|Z^{η}−Z^{ηT^{κ}}|_{T^{λ}}^{⋆ℓ} ∥_{L^{p}}^{ρ}dλ)^{1/ρ}

by induction hyp.: ≤ (ℓ+1)C_{p}^{⋄} · C_{ℓ}∥g∥_{L^{p}} max_{ρ=1,2} (∫_{κ}^{μ}(λ − κ)^{ℓρ/2} dλ)^{1/ρ}

= (ℓ+1)C_{p}^{⋄} · C_{ℓ}∥g∥_{L^{p}} max_{ρ=1,2} ·\frac{1}{(ℓρ/2 + 1)^{1/ρ}} (μ − κ)^{ℓ/2+1/ρ}

as μ − κ ≤ 1:  ≤ \sqrt{ℓ+1}C_{p}^{⋄}C_{ℓ}(μ − κ)^{(ℓ+1)/2} · ∥g∥

to ∥g_{η}|Z^{η}−Z^{ηT^{κ}}|_{T^{λ}}^{⋆ℓ+1} ∥_{L^{p}}. The contribution of the second term (C.2) can be estimated by

\frac{(ℓ+1)ℓ}{2} \frac{C_{ℓ−1}}{(ℓ−1)/2 + 1} · ∥g∥_{L^{p}} = ℓC_{ℓ−1} · ∥g∥_{L^{p}} ,

due to inequality (C.1). This establishes the claim (i) and gives the recurrence relation

C_{0} = 1 , C_{1} ≤ C_{p}^{⋄} , C_{ℓ+1} ≤ C_{ℓ} · C_{p}^{⋄}\sqrt{2(ℓ+2)} + C_{ℓ−1} · ℓ for ℓ≥2 .

4.5.21 Set Q_{1} =^{def} 𝔼[ϕ(A) · ψ(\frac{1}{A})]. Then

𝔼[ϕ(Y) · ψ(\frac{1}{A})] = 𝔼[ϕ(Y) · ψ(\frac{1}{A})[Y ≤ A]] + 𝔼[ϕ(Y) · ψ(\frac{1}{A})[Y > A]]

≤ Q_{1} + 𝔼[(ϕ(Y) − ϕ(A)) · ψ(\frac{1}{A})[Y > A]]

= Q_{1} + ∫ [A < y ≤ Y] dϕ(y) [A ≤ \frac{1}{a}] dψ(a) dℙ

≤ Q_{1} + ∫ ℙ[Y ≥ y, A ≤ y ∧ \frac{1}{a}] dϕ(y) dψ(a)

≤ Q_{1} + ∫ \frac{1}{y} · (A∧y∧\frac{1}{a}) dϕ(y) dψ(a) dℙ .

Now

q_{2} =^{def} ∫ \frac{1}{y} · (A∧y∧\frac{1}{a}) dϕ(y) dψ(a)

= ∫ \frac{1}{y} · y∧\frac{1}{a} · [A > \frac{1}{a}] dϕ(y) dψ(a) + ∫ \frac{1}{y} · A∧y · [A ≤ \frac{1}{a}] dϕ(y) dψ(a)

= ∫ [y ≤ \frac{1}{a} , a > \frac{1}{A}] dϕ(y) dψ(a) + ∫ \frac{1}{ya} · [y > \frac{1}{a} , a > \frac{1}{A}] dϕ(y) dψ(a)

+ ∫ \frac{A}{y} · [A < y , a ≤ \frac{1}{A}] dϕ(y) dψ(a) + ∫ [y < A , a ≤ \frac{1}{A}] dϕ(y) dψ(a)

≤ ∫ ∞_{\frac{1}{A}} ϕ(1/a) dψ(a) + ∫ ∞_{\frac{1}{A}} (\frac{1}{a} ∫_{1/a}^{∞} \frac{dϕ(y)}{y}) dψ(a)

+ A ∫_{A}^{∞} \frac{dϕ(y)}{y} ψ(\frac{1}{A}) + ϕ(A) · ψ(\frac{1}{A})

= ∫ ∞_{\frac{1}{A}} Φ(1/a) dψ(a) + Φ(A)ψ(\frac{1}{A}) .

Inserting this into the previous inequality results in inequality (4.5.32). An easy manipulation shows that ϕ(y) = y^{β} has Φ(y) = y^{β}/(1 − β) and that, with ψ(a) = a^{α},

∫ ∞_{\frac{1}{A}} Φ(1/a) dψ(a) = \frac{α}{(1 − β)(β − α)} · A^{β−α} .

Consequently ϕ(Y) · ψ(\frac{1}{A}) = A^{α−β} and

(Φ(A) + ϕ(A)) · ψ(\frac{1}{A}) + ∫ ∞_{\frac{1}{A}} Φ(1/a) dψ(a)

= (1 + \frac{1}{1 − β} + \frac{α}{(1 − β)(β − α)}) · A^{β−α}

= \frac{(2 − β)(β − α) + α}{(1 − β)(β − α)} · A^{β−α} ≤ \frac{2}{(1 − β)(β − α)} · A^{β−α} .

Taking expectations results in inequality (4.5.33). 4.5.26 The underlying filtration F. is assumed right-continuous. A time transformation is an increasing collection T. =^{def} {T^{λ} : 0 ≤ λ < ∞} of stopping times with T^{λ} ⟶_{λ→∞} ∞. T. is called left-continuous or rightcontinuous provided it equals

T^{.−} : λ ↦ T^{λ−} =^{def} lim{T^{κ} : λ > κ ↑ λ}
or  T^{.+} : λ ↦ T^{λ+} =^{def} lim{T^{μ} : λ < μ ↓ λ} ,

respectively. With the three associated time transformations (see Exercises 1.3.15 and 1.3.30) T^{.−}, T. , T^{.+} come the filtrations F_{T^{.−}}, F_{T.} , F_{T^{.+}} . Clearly

T^{λ−} ≤ T^{λ} ≤ T^{λ+} and F_{T^{λ−}} ⊆ F_{T^{λ}} ⊆ F_{T^{λ+}} ,  λ ≥ 0 .

It is the last filtration, \underline{F}. =^{def} F_{T^{.+}} = \{F_{T^{λ+}} : 0 ≤ λ < ∞\}, that we single out, for its right-continuity (ibidem). By interpreting λ as a new time parameter, the time transformations T^{.−}, T. , and T^{.+} can also be thought of as \underline{F}.-adapted increasing processes, T^{.−} left-continuous, and T^{.+} right-continuous. The finite random variable

Λ_{t} =^{def} inf{λ : T^{λ−} > t} = inf{λ : T^{λ} > t} = inf{λ : T^{λ+} > t}  (C.3)

is, for every t ≥ 0, an \underline{F}.-stopping time and defines a right-continuous time transformation Λ. on \underline{F}. (ibidem). Considered as a process, t ↦ Λ_{t} is easily seen to be F.-adapted. Indeed, for any μ > 0 we have [Λ_{t} < μ]

Figure C.19 A Time Transformation

= ⋃{[T^{λ+} > t] : λ < μ} = ⋃{[T^{q} > t] : ℚ ∋ q < μ} ∈ F_{t}. Its left-continuous version is at t > 0 given by

Λ_{t−} = inf{λ : T^{λ+} ≥ t} .  (C.4)

Lemma C.1 (i) Equalities (C.3) and (C.4) hold. (ii) For λ, t ≥ 0

T^{λ−} = inf{t : Λ_{t} ≥ λ} ≤ T^{λ} ≤ T^{λ+} = inf{t : Λ_{t} > λ} ;  (C.5)
[T^{λ−} ≤ t] = [λ ≤ Λ_{t}] and [Λ_{t−} ≤ λ] = [t ≤ T^{λ+}] ;  (C.6)
and thus  T^{λ−} ≤ t ≤ T^{λ+} ⟺ Λ_{t−} ≤ λ ≤ Λ_{t} .  (C.7)

(iiia) The following are equivalent: Λ. is strictly increasing; for all λ ≥ 0, T^{λ−} = T^{λ} = T^{λ+} ; one of T^{.−}, T. , T^{.+} is continuous; all of them are. (iiib) T. is strictly increasing if and only if Λ. is continuous. (iv) The T^{λ} are finite (everywhere, nearly, almost surely) if and only if Λ_{t} ⟶_{t→∞} ∞ (everywhere, nearly, almost surely). The T^{λ} are bounded if and only if inf{Λ_{t}(ω) : ω ∈ Ω} ⟶_{t→∞} ∞.

(v) If T is an F.-stopping time, then Λ_{T} is an \underline{F}.-stopping time; if L is an \underline{F}.-stopping time, then T^{L+} = T_{⟵}^{L} is an F.-stopping time.

(vi)  Λ_{∞} =^{def} sup\{λ : T^{λ} < ∞\} is an \underline{F}.-stopping time.

(vii) If the T^{λ} are nearly finite, then F. and F_{T.} have the same nearly empty sets.

Proof. (i) If inf{λ : T^{λ+} > t} < μ, then T^{λ} > t for some λ < μ and thus T^{μ−} > t and inf{λ : T^{λ−} > t} ≤ μ: inf{λ : T^{λ−} > t} ≤ inf{λ : T^{λ+} > t} follows, and with the reverse inequality being obvious we get equality throughout (C.3). As to (C.4), both sides of the equation define leftcontinuous functions of t that agree unless the level set [T^{.+} = t] has strictly positive length, which can happen only countably often. (ii) The inequalities in (C.5) are obvious. The equalities follow directly from the right-continuity of Λ. and T^{.+} in conjunction with (C.3) and (C.4). Equation (C.7) is but a summary of (C.6). (iii) Clearly T^{.−} is left-continuous and T^{.+} is rightcontinuous; they agree iff they are continuous. The equalities in (C.5) make it obvious that they do agree iff Λ. has no level sets [Λ. = t] of strictly positive length, i.e., iff Λ. is strictly increasing. (v) If Λ takes countably many values λ_{i}, then [T^{Λ} ≤ t] = ⋃[T^{Λ} ≤ t, Λ = λ_{i}] = ⋃[T^{λ_{i}} ≤ t] ∩ [Λ = λ_{i}] belongs to F_{t} inasmuch as [T^{λ_{i}} ≤ t] ∈ F_{t} and [Λ = λ_{i}] ∈ F_{T^{λ_{i}}} . In the general case use the stopping times Λ^{(n)} of exercise 1.3.20 and the right-continuity of both T_{⟵}. and of F_{⟵}_{T.} . The same argument shows that Λ_{T} is an \underline{F}.-stopping time. (vi) [λ < Λ_{∞}] = [T^{λ+} < ∞] ∈ \underline{F}_{λ}. (vii) Use equation (C.5) and exercise 3.5.19. Let now \underline{B} denote a copy of the base space B = [0, ∞) × Ω and denote its typical point by (λ, ω). Since T^{λ−} may well be infinite, we need to augment B temporarily: B′ is the base space with the graph [[∞]] = {∞}×Ω of the infinite stopping time adjoined:

B′ =^{def} [0, ∞] × Ω = B ∪ [[∞]] .

The left-continuous time transformations T^{.−} and Λ_{.−} give rise to maps (see figure C.20)

T^{.−} : \underline{B} → B′ via (λ, ω) ↦ (T^{λ−}(ω), ω)
and  Λ_{.−} : B → \underline{B}′ via (t, ω) ↦ (Λ_{t−}(ω), ω) ,

in which the image of [[0, Λ_{∞})) lies in B and that of B in [[0, Λ_{∞})), respectively. If both Λ. and T^{.−} are continuous or are strictly increasing, then T^{.−} and Λ_{.−} are inverses of each other.

Figure C.20 A Time Transformation

Let us denote by an underscore the composition with T^{.−}. To be quite precise, for X : B → ℝ,

\underline{X}_{λ}(ω) =^{def} \{ \begin{matrix}X_{T^{λ−}}(ω) & where T^{λ−}(ω) < ∞, i.e., on [[0, Λ_{∞})) \\ 0 & on [[Λ_{∞}, ∞)), i.e., for Λ_{∞}(ω) ≤ λ < ∞. \end{matrix}

If X is progressively measurable for F. , then \underline{X} is adapted to \underline{F}. (proposition 1.3.9); if X is left-continuous, then clearly so is \underline{X}. The usual sequential closure argument shows that X ↦ \underline{X} takes F.-predictable processes to \underline{F}.-predictable processes that vanish on [[Λ_{∞}, ∞)).

Definition C.2 An L^{p}-integrator Z is called compatible with the time transformation T. if (a) the stopped process Z^{T^{μ+}} is I^{p}-bounded for all μ < ∞ and (b) Z is nearly constant in time on every interval of the form [[T^{λ−}, T^{λ+}]] – by lemma C.1 (iii) this holds in particular when Λ. is strictly increasing.

Exercise C.3 If the T^{λ−} are predictable, as is typical, then the compatibility simply means that ⋃_{λ>0} [[T^{λ−}, T^{λ+}]] is Z-negligible. Proposition C.4 Let 0 ≤ p < ∞ and suppose the L^{p}-integrator Z is compatible with the time transformation T. . Then \underline{Z} is nearly right-continuous and adapted to \underline{F}. ; in fact, it is an L^{p}-integrator on \underline{F}. and satisfies

\underline{Z}^{μ} _{I^{p}} ≤ Z^{T^{μ}} _{I^{p}} .  (C.8)

Moreover, for every Z−p-integrable process X the process \underline{X} is \underline{Z}−p-integrable and

∫_{B} X_{s} dZ_{s} = ∫\underline{B} \underline{X}_{λ} d\underline{Z}_{λ} .  (C.9)

Proof. For every λ ≥ 0 let N_{λ} be the nearly empty set of ω ∈ Ω so that t ↦ Z_{t}(ω) is not constant for T^{λ−}(ω) ≤ t ≤ T^{λ+}(ω). The representation

N =^{def} ⋃_{λ} N_{λ} = ⋃_{n} ( ⋃_{λ} N^{λ} ∩ [T^{λ+} > T^{λ−} + 1/n < ∞])

exhibits their union N as a countable union of nearly empty sets, which is therefore nearly empty. Indeed, there are at most countably many λ’s for which the sets in the inner union are non-void. Upon removal of N we are left with a process Z whose paths are càdlàg and constant in time on every interval of the form [[T^{λ−}, T^{λ+}]] not only nearly but in fact everywhere. If

λ_{n} ↓ λ, then \underline{Z}_{λ_{n}} = Z_{T^{λn−}} = Z_{T^{λn+}} ⟶_{n→∞} Z_{T^{λ+}} = Z_{T^{λ−}} = \underline{Z}_{λ}, therefore \underline{Z} is

right-continuous at any λ ≥ 0.

Let then  X =^{def} f_{0}[[0]] + ∑_{n=1}^{N} f_{n}·((λ_{n}, λ_{n+1}]] ,  f_{n} ∈ L^{∞}(\underline{F}_{λ_{n}}) ,

be a typical elementary integrand from E[\underline{F}.] with λ_{N+1} ≤ μ (see (2.1.1)).

Then ∫ X d\underline{Z} = f_{0}·Z_{0} + ∑_{n=1}^{N}f_{n}·(Z_{T^{λ_{n+1}}} − Z_{T^{λn}} ) = ∫ X′ dZ ,
where  X′ =^{def} f_{0}·[[0]] + ∑_{n=1}^{N}f_{n}·((T^{λ_{n}}, T^{λ_{n+1}}]] ∈ P[F.]

From this equation (C.8) is evident. Let X = f · ((s, t]] with f ∈ F_{s}. Then^{3}

\underline{((s, t]]}_{λ} = [s < T^{λ−} ≤ t] = [Λ_{s} < λ ≤ Λ_{t}] = ((Λ_{s}, Λ_{t}]]_{λ}

3 In accordance with convention A.1.5 on page 364 sets are identified with their (idempotent) indicator functions. A stochastic interval ((S, T]], for instance, has at the instant s the value ((S, T]]_{s} = [S < s ≤ T] = \{ \begin{matrix}1 & if S(ω) < s ≤ T (ω) \\ 0 & elsewhere \end{matrix} .

and  ∫_{B} X dZ = ∫ f · ((s, t]] dZ = f · (Z_{t} − Z_{s})
as T^{Λ_{t}−} ≤ t ≤ T^{Λ_{t}}: = f · (Z_{T^{Λ_{t}}} − Z_{T^{Λs}} ) = ∫ f · ((Λ_{s}, Λ_{t}]] d\underline{Z}
= ∫ \underline{X}_{λ} d\underline{Z}_{λ} .

By linearity, (C.9) is true for X ∈ E, and then for bounded predictable X. It is a matter of bookkeeping to extend this to Z−0-integrable processes X.

C.5 An integrator Z compatible with T. has \underline{[Z, Z]} = [\underline{Z}, \underline{Z}].

Proof. For 0 ≤ μ < ∞

∫_{0+}^{T^{μ−}} Z_{.−} dZ = ∫_{B} ((0, T^{μ−}]] · Z_{.−} dZ = ∫_{[[0,Λ_{∞}))} ((0, T^{μ−}]]_{T^{λ−}} · \underline{Z}_{λ−} d\underline{Z}_{λ}
= ∫ ((0, T^{μ+}]]_{T^{λ−}} \underline{Z}_{λ−} d\underline{Z}_{λ} = ∫_{0+}^{μ} \underline{Z}_{λ−} d\underline{Z}_{λ} .

Thus \underline{[Z, Z]}_{μ} = [Z, Z]_{T^{μ−}} = Z_{T^{μ−}}^{2} − Z_{0}^{2} − 2 ∫_{0+}^{T^{μ−}} Z_{.−} dZ
= \underline{Z}_{μ}^{2} − \underline{Z}_{0}^{2} − 2 ∫_{0+}^{μ} \underline{Z}_{λ−} d\underline{Z}_{λ} = [\underline{Z}, \underline{Z}]_{μ} .

C.6 Let M be a continuous local martingale on (Ω, F, ℙ), set Λ = [M, M], and introduce the time transformations T^{λ−}, T^{λ+} of (C.5), setting T^{λ} =^{def} T^{λ−}. By inequality (4.2.8), M is constant on the intervals [[T^{λ−}, T^{λ+}]], so \underline{M} is a continuous local martingale. If Λ_{t} ⟶_{t→∞} ∞, then by item C.5 and corollary 3.9.5, \underline{M} is a standard Wiener process on \underline{F}. . If ℙ[Λ_{∞} < ∞] > 0, let W be a Wiener process on (Ω′, F.^{′}, ℙ′) that is independent of (F_{∞}, ℙ), and set \underline{M}′ =^{def} [[0, Λ_{∞}))∗\underline{M} + [[Λ_{∞}, ∞))∗W. Show that \underline{M}′ is a standard Wiener process on (Ω×Ω′, F.×F.^{′}, ℙ×ℙ′).

Sure Control of Integrators can be had in a manner similar to item C.6. Suppose Z is a vector of L^{q}-integrators, q ≥ 2, and Λ = Λ^{⟨q⟩}[Z] is its previsible controller from theorem 4.5.1. It is nary a loss of generality to assume that Λ is strictly increasing and Λ_{t} ⟶_{t→∞} ∞ (see remark 4.5.2). By lemma C.1, we then have equality of continuous time transformations

T^{λ−} =^{def} inf{t : Λ_{t} ≥ λ} = T^{λ} =^{def} T^{λ+} =^{def} inf{t : Λ_{t} > λ} .

Since Λ_{∞} = ∞, the map T^{.−} : \underline{B} → B is continuous and surjective. The controlling estimate (4.5.1) turns into

∥|\underline{X}∗\underline{Z}|_{Λ}^{⋆}∥_{L^{p}} ≤ C_{p}^{⋄} · max_{ρ=1^{⋄},p^{⋄}} ∥(∫_{0}^{Λ+} |\underline{X}|_{λ}^{ρ} dλ)^{1/ρ}∥_{L^{p}}

for any \underline{F}.-stopping time Λ and any p ∈ [2, q]. Continue on. 4.6.5 (adapt exercise 1.3.47 or theorem 5.7.3 (iii)). 4.6.14 (i) Equation (4.6.19) has the easy consequence

𝔼[exp (i ∫ h dȷ_{Z})] = 𝔼[exp (i ∑_{j} α_{j}ȷ_{Z}(A_{j}))]

= ∏_{j} exp ((ν×λ)(A_{j})(e^{iα_{j}} − 1)) ,

showing that the random variables ȷ_{Z}(A_{j}) are independent Poisson with means (ν×λ)(A_{j}). 4.6.21 A gives rise to a Lévy process Z (page 267). The definition (4.6.31) produces a Feller semigroup whose generator is necessarily dissipative and conservative and is given by A (equation (4.6.32)) on S. S is dense in C_{0} and invariant under both T.^{D} and T.^{J} , thus under T. , and therefore is a core for A (exercise A.9.4). 5.1.7 (i) The function f(x) =^{def} ∫_{0}^{|x|} s ∧ 1 ds of example A.2.48 serves again: the paths e./n converge to zero in 𝔰_{1} , yet the remainder

RF (e./n; 0) = f(e./n) − f(0) − f′(0)·e./n = f(e./n)
has  ∥RF (e./n; 0)∥_{1} = ∥e./n∥_{1} ≠ o(∥e./n∥_{1}) .

(ii) On the positive side, if M^{∘} > M, pick n ∈ ℕ and let ϵ > 0 be given. There exists a t > n so that 2Le^{(M−M^{∘})t} ≤ ϵ. There exists a δ > 0 so that for all u, v ∈ U in the ball of radius n and all x, y ∈ ℝ^{n} in the ball of radius ne^{M^{∘}t}, |v−u| + |y−x| ≤ δ implies Df(v, y) − Df(u, x) ≤ ϵ. Let |v−u| + ∥y. − x. ∥ ≤ δe^{−Mt}. Then |y_{s} − x_{s}| ≤ δ for s ≤ t and so

|Rf(v, y_{s}; u, x_{s}]| ≤ e^{Ms} ∫_{0}^{1} Df((u, x_{s}) + λ(v−u, y_{s}−x_{s})) − Df(u, x_{s}) dλ
× ∥y. − x.∥_{M}

≤ \{ \begin{matrix}ϵe^{Ms} & for s ≤ t \\ e^{Ms}2L ≤ ϵe^{M^{∘}s} & for s > t \end{matrix}\} × ∥y. − x.∥_{M} ,

which implies \begin{matrix}∥Rf[v, y.; u, x.]∥ & M^{∘} \\ ∥y. − x.∥ & M \end{matrix} ≤ ϵ .

5.1.8 (i) Both ξ_{.+s}^{f}(x) and ξ.^{f}(ξ_{s}^{f}(x)) satisfy dX_{t} = f(X_{t}) dt with X_{0} = ξ_{s}^{f}(x). (ii) Hint: Define Dξ_{t}^{f}[x] as the solution of equation (5.1.24) on page 278. Then write a differential equation for Δ. =^{def} ξ.^{f}(x) − ξ.^{f}(x′) − Dξ.^{f}[x] · (x−x′). Then show, using inequality (5.1.16) on page 276, that ∥Δ. ∥/|x−x′| ⟶_{x−x′→0} 0. This means that Dξ.^{f}[x] is the Fréchet derivative at x of ξ.^{f} : x′ ↦ ξ.(x′), map from ℝ^{n} to 𝔰 (see definition A.2.49 on page 390). As for equation (5.1.24), both sides answer the same initial value problem. 5.1.10 (ii) By the chain rule, σ ↦ Ξ^{f} [x, z_{σ}] solves (5.1.25). 5.1.11 (i) The exponential limit for the growth in time follows from item 5.1.4. (ii) Set ^{0}ξ′[x, t] =^{def} ξ′[x, t] − x and 0 ≤ s ≤ δ. Since ^{0}ξ′[c, 0] = 0 ∀ c ∈ ℝ^{n}, we have ξ′_{;ν}[c, 0] = 0 for all c ∈ ℝ^{n} and, for some c_{s} between c, c′,

0ξ′[c′, s] −^{0}ξ′[c, s] = ^{0}ξ′_{;ν}[c_{s}, s](c′−c)^{ν}
= (^{0}ξ′_{;ν}[c_{s}, s] −^{0}ξ′_{;ν}[c_{s}, 0])(c′−c)^{ν} ,
whence  |^{0}ξ′[c′, s] −^{0}ξ′[c, s]| ≤ L′δ · |c′−c| ≤ (e^{L′δ} − 1) · |c′−c|
and  |ξ′[c′, s] − ξ′[c, s]| ≤ e^{L′δ} · |c′−c| ,  0 ≤ s ≤ δ .

(iii) For fixed t and δ set k =^{def} ⌈t/δ⌉ and t_{i} =^{def} iδ for i = 0, 1, …, k. Then t_{k−1} < t ≤ t_{k}. Let Δ_{i}^{⋆} denote the maximal function of the difference of the global solution at t_{i}, which is x_{t_{i}} = ξ[c, t_{i}], from its ξ′-approximate x′_{t_{i}}. Consider an s ∈ [t_{i}, t_{i+1}].

Since  x′_{s} − x_{s} = ξ′[x′_{t_{i}}, s−t_{i}] − ξ′[x_{t_{i}}, s−t_{i}]
+ ξ′[x_{t_{i}}, s−t_{i}] − ξ[x_{t_{i}}, s−t_{i}] ,
we have  |Δ_{i+1}^{⋆}| ≤ |Δ_{i}^{⋆}| × e^{L′δ} + (|x|_{t_{i}}^{⋆}+1) × (\underline{m}δ)^{r}e^{\underline{m}δ} ,
which implies  |Δ_{k}^{⋆}| ≤ (|x|_{t_{k}}^{⋆}+1) × (\underline{m}δ)^{r}e^{\underline{m}δ} · ∑_{0≤i<k} e^{iL′δ}

for x. ∈ 𝔰_{M}, as t_{k} = kδ: ≤ (∥x.∥_{M}e^{Mt_{k}} + 1) × (\underline{m}δ)^{r}e^{\underline{m}δ} · \frac{e^{L′kδ} − 1}{e^{L′δ} − 1}
by (5.1.16), as δ → 0:  ≤ \frac{2}{1−γ} (∥c∥_{M}+\frac{|f(0)|}{eM}+1) e^{Mt_{k}} × (\underline{m}δ)^{r}e^{\underline{m}δ}·k e^{L′t_{k}}
since k = t_{k}/δ:  ≤ const (∥c∥_{M} + 1)\underline{m}^{r}e^{\underline{m}δ}δ^{r−1} × t_{k}·e^{(M+L′)t_{k}}
≤ \overline{b}·(|c|+1) × δ^{r−1}·e^{\overline{m}t_{k}}

for suitable \overline{b} = \overline{b}[f; ξ′] and \overline{m} = \overline{m}[f; ξ′] > M+L′. 5.2.3 We know already from inequality (5.2.6) that

∥(F^{ν}∗Z)_{T^{μ}−}^{⋆}∥_{L^{p}}^{∗} ≤ C_{p}^{⋄} max_{ρ=1,p} ∥( ∫_{0}^{μ} |F_{T^{λ}−}^{ν}|_{∞}^{ρ} dλ)^{1/ρ}∥_{L^{p}}^{∗}
by exercise A.3.29:  ≤ C_{p}^{⋄} max_{ρ=1,p} ( ∫_{0}^{μ} ∥|F_{T^{λ}−}^{ν⋆}|_{∞}∥_{L^{p}}^{∗ρ} dλ)^{1/ρ} .

Taking the p-norm for counting measure and using Fubini’s theorem gives

∥|(F ∗Z)_{T^{μ}−}^{⋆}|_{p}∥_{L^{p}}^{∗} ≤ C_{p}^{⋄} max_{ρ=1,p} ( ∫_{0}^{μ} ∥|F_{T^{λ}−}^{⋆}|_{∞p}∥_{L^{p}}^{∗ρ} dλ)^{1/ρ}

Measuring in L^{p}(Me^{−Mμ} dμ) the part that appears for ρ = p on the righthand side gives a number less than M^{−1/p} · |F |_{∞} ._{p,M}. For the case ρ = 1 we set f(λ) =^{def} ∥|F_{T^{λ}−}|_{∞p}^{⋆} ∥_{L^{p}}^{∗ρ} and α =^{def} M/(p + p′) and estimate

∫_{0}^{μ} f(λ) dλ = ∫ [λ ≤ μ]e^{αλ} · [λ ≤ μ]f(λ)e^{−αλ}dλ

≤ (∫_{0}^{μ} e^{αp′λ}dλ)^{1/p′}·(∫_{0}^{μ} f(λ)^{p}e^{−αpλ}dλ)^{1/p}

< (\frac{1}{αp′} )^{1/p′}e^{αμ} · (∫ [λ ≤ μ]f(λ)^{p}e^{−αpλ} dλ)^{1/p}

whence

∫ (∫_{0}^{μ} f(λ) dλ)^{p}Me^{−Mμ}dμ < (\frac{1}{αp′} )^{\frac{p}{p′}} · ∫ ∫ [λ ≤ μ]f(λ)^{p}e^{−αpλ}Me^{(αp−M)μ}dμdλ

< (\frac{1}{αp′} )^{p/p′} · ∫ f(λ)^{p}e^{−αpλ} \frac{M}{αp − M} e^{(αp−M)μ}|_{λ}^{∞} dλ

= (\frac{1}{αp′} )^{p/p′} \frac{1}{M − αp} · ∫_{0}^{∞} f(λ)^{p} Me^{−Mλ} dλ

≤ (\frac{1}{αp′} )^{p/p′} \frac{1}{M − αp} · |F |_{∞} _{p,M}^{.p}

= (\frac{p}{M} )^{p} · |F |_{∞} _{p,M}^{.p}

Thus  F_{.−}∗Z ._{p,M} ≤ C_{p}^{⋄}(\frac{1}{M^{1/p}} ∨ \frac{p}{M} ) · |F |_{∞} _{p,M}^{.p} .

5.2.16 needed 5.2.17 First consider the equation X = 1 + X∗W, whose solution is the Doléans–Dade exponential ℰ_{t} = e^{W_{t}−t/2} of W (see proposition 3.9.2 on page 159). Here Λ_{t}^{⟨q⟩}[W] = t ∀ q, and by exercise 4.5.6 on page 240, dΛ_{t}^{⟨q⟩}[ℰ] = e^{qW_{t}−qt/2} dt. Now if Λ_{t}^{⟨q⟩}[ℰ] were dominated by a sure controller ξ, i.e., dΛ_{t}^{⟨q⟩}[ℰ] ≤ dξ, then the absolutely continuous part dξ^{∥} of dξ would have to have a locally integrable Radon–Nikodym Derivative h(t) = dξ^{∥}(t)/dt < ∞, which would have to satisfy e^{qW_{t}−qt/2} ≤ h(t) almost surely; since ℙ[W_{t} > K] > 0 for all K ∈ ℕ and t > 0, this is impossible. We see that some boundedness assumption on the value ^{0}F [X] of ^{0}F at the solution X of theorem 5.2.15 on page 291 is needed.

Assume then that Λ^{⟨q⟩} is dominated by the sure controller η: dΛ^{⟨q⟩}[Z] ≤ dη, and that ^{0}F [X] is a bounded process. By exercise 4.5.6 on page 240, the controllers Λ^{⟨q⟩}[X^{η}] of the components X^{η} of X satisfy dΛ^{⟨q⟩}[X^{η}] ≤ const × dη, and then so does dΛ^{⟨q⟩}[X]. 5.2.20 needed 5.2.21 ^{0}𝔘[X] =^{def} 𝔘[X] − C has the same contractivity modulus γ < 1 as 𝔘. Thus C _{p,M}^{⋆} = X −^{0}𝔘[X] _{p,M}^{⋆} ≤ X _{p,M}^{⋆} + ^{0}𝔘[X] −^{0}𝔘[0] _{p,M}^{⋆} ≤ (1 + γ) X _{p,M}^{⋆}.

5.2.22 Apply (5.2.34) with C = C[u], C′ = C[v] and F [ . ] = F [u, . ], F′[ . ] = F [v, . ].

5.2.24 : Let t < ∞. Z^{t} is an L^{p}-integrator for some p > dim U and an equivalent probability ℙ′. The assumed Lipschitz conditions imply that the stopped processes X[u]^{t} satisfy (5.2.41) for ℙ′ (proposition 5.2.22). Corollary 5.2.23 shows that they are nearly continuous in u ∈ U. Then let t → ∞.

5.3.11 (i) Let A be a symmetric bilinear form on euclidean space ℝ^{n}. Then sup{|A(ξ, η)| : |ξ |_{2} ≤ 1, |η |_{2} ≤ 1} = sup{|A(ξ, ξ)| : |ξ |_{2} ≤ 1}. This is easily seen by diagonalizing the symmetric matrix A; in fact, |A(ξ, η)| is strictly less than the supremum above unless ξ and η are collinear and of unit euclidean length. (ii) Let now A be a symmetric k-linear form on euclidean space ℝ^{n}. Then sup{|A(ξ_{1}, …, ξ_{k})| : |ξ_{i} |_{2} ≤ 1, 1≤i≤k} is taken at a k-tuple (ξ_{1}, …, ξ_{k}) of unit vectors. Considering A(ξ_{1}, ξ_{2}, …) a bilinear form in (ξ_{1}, ξ_{2}) and using (i) shows that ξ_{1} = ±ξ_{2} . Similarly ξ_{i} = ±ξ_{j} for 1 ≤ i < j ≤ k. Thus the supremum is taken also at a k-tuple of the form (ξ, ξ, …, ξ). (iii) Next let D^{∗} be a k-linear scalar form on a seminormed space (E, ∥ ∥_{E}). Let x_{i} ∈ E_{1} , i = 1, …, k, with a < ∥D^{∗}(x_{1}, …, x_{k})∥_{S}. Define the k-linear form A on euclidean space ℝ^{k}

by A(ξ_{1}, …, ξ_{k}) =^{def} D^{∗}( ∑ ξ_{1}^{κ}x_{κ}, …, ∑ ξ_{k}^{κ}x_{κ}). By (ii) there is a ξ ∈ ℝ^{k} so

that, with x =^{def} ∑ ξ^{κ}x_{κ}, a < D^{∗}(x, …, x). Now ∥x∥_{E} ≤ ∑_{κ} |ξ_{κ}| ≤ k^{1/2} .

Thus a < k^{k/2}{sup D^{∗}(x, …, x) : ∥x∥_{E} ≤ 1}. (iv) Finally let D be a k-linear map from a seminormed space (E, ∥ ∥_{E}) to another seminormed space (S, ∥ ∥_{S}). Let a < sup{∥D(x_{1}, …, x_{k})∥_{S} : ∥x_{i} ∥_{E} ≤ 1}. There are a linear form y^{∗} in the dual S^{∗} that has norm ∥y^{∗} ∥_{S^{∗}} ≤ 1 and elements

x_{1}, …, x_{k} in E so that D^{∗} =^{def} y^{∗} ∘ D has a < ∥D^{∗}(x_{1}, …, x_{k})∥_{S}. By (iii),

there is an x ∈ E_{1} with a < k^{k/2}D^{∗}(x, …, x):

sup{∥D(x_{1}, …, x_{k})∥_{S} : ∥x_{i}∥_{E} ≤ 1}≤ k^{k/2} sup{∥D(x, …, x)∥_{S} : ∥x∥_{E} ≤ 1} .

5.3.18 Let us pick a u ∈ U, and write D^{λ} for D^{λ}F [u] and T^{λ}[v] for T^{λ}F [u](v). For v, v′ ∈ U

T^{l}[v′] − T^{l}[v] = ∑_{0≤λ≤l} \frac{D^{λ}}{λ!} · [(v′−u)^{⊗λ} − (v−u)^{⊗λ}]

= ∑_{0≤λ≤l} \frac{D^{λ}}{λ!} · [ ∑_{0≤i≤λ} \binom{λ}{i}(v−u)^{⊗λ−i} ⊗ (v′−v)^{⊗i} − (v−u)^{⊗λ}]

= ∑_{0≤λ≤l} \frac{D^{λ}}{λ!} · ∑_{0<i≤λ} \binom{λ}{i}(v−u)^{⊗λ−i} ⊗ (v′−v)^{⊗i}

= ∑_{0<i≤l} ∑_{i≤λ≤l} \frac{\binom{λ}{i}D^{λ}}{λ!} · (v−u)^{⊗λ−i} ⊗ (v′−v)^{⊗i} ,

whence

D^{i}T^{l}[v]·ξ^{⊗i} = ∑_{i≤λ≤l} \frac{i!\binom{λ}{i}D^{λ}}{λ!} · (v−u)^{⊗λ−i} ⊗ ξ^{⊗i}  , i = 1, ⋯, l

= D^{i}F [u]·ξ^{⊗i} + ∑_{i<λ≤l} \frac{D^{λ}}{(λ − i)!} · (v−u)^{⊗λ−i}⊗ ξ^{⊗i}

= D^{i}T^{l}[u]·ξ^{⊗i} + D^{i+1} · ξ^{⊗i}⊗(v−u) + r(v−u) ,

where each summand of r(v−u) contains a power (v−u)^{⊗p} with p ≥ 2. Hence v ↦ D^{i}T^{l}F [u](v)·ξ^{⊗i} has ith derivative η ↦ D^{i+1}F [u] · ξ^{⊗i} ⊗η at v = u, for i = 1, …, l − 1. In fact, since the derivatives appearing in r(v−u) are bounded, v ↦ D^{i}T^{l}F [u](v)·ξ^{⊗i} is uniformly differentiable. Let us define G[v] =^{def} F [v] − T^{l}F [u](v). This function is l-times weakly uniformly differentiable with D^{0}G[u] = … = D^{l}G[u] = 0 and ∥G[u]∥_{S}^{∘} = o(∥v−u∥_{E}^{l}). It is left to prove that v ↦ D^{i}G[v]·ξ^{⊗i} has vanishing derivative at u, for i = 1, …, l − 1. Let ξ, ξ′ ∈ E be unit vectors and τ ∈ ℝ and set

v =^{def} u + τξ, v′ =^{def} u + τξ′

Then both G[v′] − G[v] = (G[u + τξ]−G[u]) − (G[u + τξ′]−G[u])

= ∑_{0<i≤l} τ^{i}D^{i}G[v] · (ξ′ − ξ)^{⊗i} + R^{l}G[v; v′]

and R^{l}G[v; v′] are o(τ^{l}) when measured with ∥ ∥_{S}^{∘}, independently of the location of u ∈ U. Hence so is the sum above. This implies

\frac{D^{i}G[τv] · (ξ′ − ξ)^{⊗i}}{τ^{l−i}} ⟶_{τ→0} 0  , i = 0, …, l ;
and therefore  D{v ↦ D^{i}G[v] · (ξ′ − ξ)^{⊗i}} = 0 at u , i = 0, …, l−1 .

5.3.19 Answer forthcoming. 5.4.8 Since the mesh of T goes to zero, we have |F′[X]−F [X]|_{∞} → 0 pointwise on H × B, after composition with the time transformation T. on {1, …, n} × {λ : 0 ≤ λ < ∞} × Ω. The Dominated Convergence Theorem produces F^{T} [X]−F [X] ._{p,M} ⟶_{δ→0} 0 for all M > 0 after integration suitable powers over dℙ and dλ, and exercise 5.2.3 on page 285 yields

F^{T} [X]−F [X] _{p,M}^{⋆} ⟶_{δ→0} 0.

5.4.12 Apply theorem 3.9.24 on page 170. 5.4.17 (i) Let P be such that |Φ_{;ν}[x, z]| ≤ P (|z|) for Φ = Ξ_{;η} and Φ = Ξ′_{;ην}Ξ_{;θ}^{′ν}. In fact, let Φ be one of these (Itô-) coefficients, say the one with index ι, and, given λ, pick k so that κ =^{def} kδ ≤ λ < (k+1)δ. Write

ζ^{T} =^{def} Z−Z^{T} . Then

∥\overline{F} ′_{ι}[Y]_{T^{λ}} − \overline{F} ′_{ι}[X]_{T^{λ}} ∥_{L^{p}} = ∥Φ[Y_{T^{κ}}, ζ_{T^{λ}}^{T^{κ}}] − Φ[X_{T^{κ}}, ζ_{T^{λ}}^{T^{κ}}]∥_{L^{p}}
by the mean value theorem:  ≤ ∥|Y − X|_{T^{κ}}^{⋆}P (|Z_{T^{λ}} − Z_{T^{κ}} |)∥_{L^{p}}
by inequality (4.5.29):  ≤ ∥|Y − X|_{T^{κ}}^{⋆} ∥_{L^{p}} × P′(\sqrt{λ−κ})
with L =^{def} P′(\sqrt{δ}):  ≤ L · ∥|Y − X|_{T^{κ}}^{⋆} ∥_{L^{p}} ≤ L · ∥|Y − X|_{T^{λ}}^{⋆} ∥_{L^{p}} .

(ii) From Ξ′[C, z] = C + Ξ′_{;η}[C, 0]z^{η} + Ξ′_{;ηθ}[C, \overline{z}]z^{η}z^{θ} and the assumption

Ξ′_{;η}, Ξ′_{;ηθ} ∈ BP, we see that ^{0}Ξ′[C, z] =^{def} Ξ′[C, z] − C has |^{0}Ξ′[C, z]| ≤ P (|z|),

P being a polynomial with zero constant term. Therefore, by (4.5.27), ∥|^{0}Ξ′[C, ζ^{T^{κ}}]|_{T^{λ}}^{⋆} ∥_{L^{p}} ≤ P′(\sqrt{λ−κ}), P′ being another polynomial with zero constant term. From this, inequality (5.4.31) is evident. Also, we see here that ∥|^{0}Ξ′[C′, ζ.^{T^{κ}}] −^{0}Ξ′[C, ζ.^{T^{κ}}]|_{T^{λ}}^{⋆} ∥_{L^{p}} is bounded by a multiple of \sqrt{λ−κ} for λ in some neighborhood of κ, whence inequality (5.4.32) for λ ≈ κ.

For large λ we write |Ξ′[C′, z] − Ξ′[C, z]| ≤ |C′ − C| · P (|z|) .

This implies

∥|Ξ′[C′, ζ.^{T^{κ}}] − Ξ′[C, ζ.^{T^{κ}}]|_{T^{λ}}^{⋆} ∥_{L^{p}} ≤ ∥C′−C∥_{L^{p}} · P′(\sqrt{λ−κ})
for some (other) L′:  ≤ ∥C′−C∥_{L^{p}} · e^{L′(λ−κ)} .

5.4.20 X−X′ solves Δ = \{(\overline{F} [X′]_{ι} − \overline{F} ′_{ι}[X′]_{ι})∗\overline{Z}^{ι}\}+(\overline{F}_{ι}[Δ + X′] − \overline{F}_{ι}[X′])∗\overline{Z}^{ι}, so by exercise 5.2.21, (\overline{F} [X′]_{ι} − \overline{F} ′_{ι}[X′]_{ι})∗\overline{Z}^{ι} _{p,M} = o(δ). By theorems 4.3.1 and 2.3.6 the martingale parts (. .)∗Z̃^{η} satisfy the same estimate; if Z happens to be a martingale, this reads (f_{η}(X′) − F′_{η}[X′])∗Z^{η} _{p,M} = o(δ). If it so happens that Z is a standard Wiener process, then it follows from theorem 4.2.12 that ∥{ ∫_{0}^{δ} (f_{η}(X′_{s}) − F′_{η}[X′]_{s})^{2}ds}^{1/2} ∥_{L^{p}} = o(δ). In particular for p = 2, ∫_{0}^{δ} ∥f_{η}(C +^{0}Ξ′(s)) −^{0}Ξ′_{;η}(s)∥_{L^{2}}^{2} ds = o(δ^{2}). Now the integrand is the square of a function ϕ that obeys |ϕ(t) − ϕ(s)| ≤ const · \sqrt{|t − s|} (see inequality (5.5.9) on page 333). If such ϕ also has ∫_{0}^{δ} ϕ^{2}(s)ds = o(δ^{2}), it must have ϕ(δ) = o(\sqrt{δ}). Else there exists an a with ϕ(t) ≥ a\sqrt{t} arbitrarily close to 0. With b the Hölder 1/2-continuity modulus, ϕ(s) ≥ a\sqrt{t} − b\sqrt{t − s} ≥ 0 for γt ≤ s ≤ t. γ = (b^{2} − a^{2})/b^{2} . Then ∫_{0}^{t} ϕ^{2}(s) ds ≥ ∫_{γt}^{t} ϕ^{2}(s) ≥ C(a, b)t^{2}. Contradiction. This is equation (5.4.36). 5.4.31 Set K =^{def} t/δ. Then the number of evaluations of ξ′ is

N = |Z_{δ}|/δ + |Z_{2δ}|/δ + … + |Z_{Kδ}|/δ
= (δ + |W_{δ}|) + (2δ + |W_{2δ}|) + … (Kδ + |W_{Kδ}|)

and has expectation

N_{1} = \frac{const}{δ}(δK(K+1)/2 + const∑_{1}^{K} \sqrt{kδ})

≈ \frac{const}{δ}(t^{2}/δ + const\sqrt{δ} · K^{3/2})

≈ \frac{b_{1}t^{2} + b_{2}t^{3/2}}{δ^{2}} = \frac{B_{1}(t)}{δ^{2}} .

By inequality (5.4.50) on page 328, δ^{−2} ≈ B_{1}^{2/r}e^{2M_{1}t/r}, whence the claim.

5.4.32 We pick an exponent p ≥ 2 and a growth constant M > M_{p,L}^{⋄(5.2.20)} to our liking, and let γ =^{def} γ_{p,M,L}^{(5.2.21)} < 1 denote the modulus of contractivity of Y ↦ 𝔘[Y] =^{def} C + \overline{f}(Y) ∗ Z in 𝔖_{p,M}^{⋆n} that goes with these choices. Having picked a δ > 0, we fix the partition by T_{k} =^{def} k · δ, set X′_{0} =^{def} c, and continue recursively: when at time T_{k} the approximate solution X′_{t} has been defined for 0 ≤ t ≤ T_{k} and the signal Z_{t} has been observed at time t ∈ [T_{k}, T_{k+1}] set^{4},

(f·(Z_{t}−Z_{t}^{T_{k}}))(x) =^{def} ∑_{η}f_{η}(x)ζ_{t}^{kη} ,
and  X′_{t} =^{def} ξ′[X′_{T_{k}}, 1; f·(Z_{t}−Z_{t}^{T_{k}})] .

The next and last item on the agenda is of course the estimation of the deviation of the exact solution X to (5.4.25) from its Ξ′-approximate X′ made with step size δ. We set the stage. X is the solution to the fixed point problem

X_{t} = C + 𝔘[X]_{t} , where 𝔘[Y] =^{def} \overline{f}_{η}(Y)∗Z^{η} ;

the Ξ′-approximate X′ is the one and only continuous process with

X′_{t} = C + 𝔘′[X′]_{t} =^{def} C + ∑_{k} ((T_{k}, T_{k+1}]]_{t} ξ′[X′_{T_{k}}, 1; f·(Z_{t}−Z_{t}^{T_{k}})] ;

and X′′ shall be the auxiliary process

X′′_{t} = C + 𝔘′′[X′′]_{t} =^{def} C + ∑_{k} ((T_{k}, T_{k+1}]]_{t} ξ[X′_{T_{k}}, 1; f·(Z_{t}−Z_{t}^{T_{k}})] .
Then Δ_{t} =^{def} X′_{t} − X_{t} = 𝔘′[X′]_{t} − 𝔘[X]_{t}
= \underbrace{𝔘′[X′]_{t}−𝔘[X′′]_{t}} + \underbrace{𝔘[X′′]_{t}−𝔘[X′]_{t}} + \underbrace{𝔘[Δ+X]_{t} − 𝔘[X]_{t}}

=  D_{t}^{(1)}  +  D_{t}^{(2)}  + ∫_{0}^{t} \overline{G}_{η}[Δ]_{s} dZ_{s}^{η} .

4 In practice one would not want to do this for all t ∈ [[T_{k}, T_{k+1}]], of course, only for t = T_{k+1} , k = 0, 1, . . .. The generality adopted is in keeping with our definition 5.4.14.

This exhibits Δ as the solution of a stochastic differential equation whose (endogenous but non-autologous) coupling coefficients \overline{G}_{η}[Δ_{t}] =^{def} \overline{f}_{η}(Δ_{t}+X_{t}) − \overline{f}_{η}(X_{t}) are Lipschitz with constant L and vanish at Δ = 0, and with nonconstant initial condition D^{(1)} + D^{(2)} . According to inequality (5.1.16) on page 276,

Δ _{p,M}^{⋆} ≤ \frac{1}{1 − γ} · ( D^{(1)} _{p,M}^{⋆} + D^{(2)} _{p,M}^{⋆} ) .

To estimate Δ it suffices therefore to estimate D^{(1)} and D^{(2)} . We start with D^{(2)} . For T_{k} ≤ s ≤ T_{k+1} we have

|X′′_{s} − X′_{s}| = |ξ[X′_{T_{k}}, 1; f·(Z_{s}−Z_{s}^{T_{k}})] − ξ′[X′_{T_{k}}, 1; f·(Z_{s}−Z_{s}^{T_{k}})]|
≤ (\underline{m}[f·(Z_{s}−Z_{s}^{T_{k}}); ξ′] · s)^{r+1} e^{\underline{m}[f·(Z_{s}−Z_{s}^{T_{k}});ξ′]·s} .
With  m =^{def} sup\{\underline{m}[f_{η}z^{η}; ξ′] : |z| ≤ 1\}  (C.10)

this gives

|X′′_{s} − X′_{s}| ≤ (m|Z_{s}−Z_{s}^{T_{k}}|)^{r+1} e^{m|Z_{s}−Z_{s}^{T_{k}} |} .  (C.11)

There is of course an assumption hidden in definition (C.10), namely

Condition C.7 If ξ′ is locally of order r + 1 on f_{1}, …, f_{d}, then \underline{m}[ . , ξ′] is bounded on their convex hull. (If, as is often the case, \underline{m}[f; ξ′] can be estimated by polynomials in the uniform bounds of various derivatives of f, then the present condition is easily verified.)

Therefore

D_{t}^{(2)} = ∫_{0}^{t} \overline{f}_{η}(X′′_{s}) − \overline{f}_{η}(X′_{s}) dZ^{η}

has size

∥|D^{(2)}|_{T_{k}}^{⋆} ∥_{L^{p}} ≤ C_{p}^{⋄(4.5.1)}d · max_{ρ=1^{⋄},2} ∥(∫_{0}^{T_{k}} |\overline{f}_{η}(X′′_{s})−\overline{f}_{η}(X′_{s})|^{ρ}ds)^{1/ρ}∥_{L^{p}}

≤ C_{p}^{⋄}dL · max_{ρ=1^{⋄},2} ∥(∫_{0}^{T_{k}} |X′′_{s} − X′_{s}|^{ρ} ds)^{1/ρ}∥_{L^{p}}

by A.3.29:  ≤ C_{p}^{⋄}dL · max_{ρ=1^{⋄},2} (∫_{0}^{T_{k}} ∥X′′_{s} − X′_{s}∥_{L^{p}}^{ρ} ds)^{1/ρ}

= C_{p}^{⋄}dL · max_{ρ=1^{⋄},2} ( ∑_{0≤κ<k} ∫_{T_{κ}}^{T_{κ+1}} ∥X′′_{s} − X′_{s}∥_{L^{p}}^{ρ} ds)^{1/ρ}

by (C.11):  ≤ C_{p}^{⋄}dL· max_{ρ=1^{⋄},2}( ∑_{0≤κ<k}∫_{T_{κ}}^{T_{κ+1}}∥(m|Z_{s}−Z_{s}^{T_{k}}|)^{r+1}e^{m|Z_{s}−Z_{s}^{T_{k}} |}∥_{L^{p}}^{ρ} ds)^{\frac{1}{ρ}}

= C_{p}^{⋄}dLm^{r+1}·max_{ρ=1^{⋄},2} k^{1/ρ} · (∫_{0}^{δ} ∥|Z_{s}|^{r+1} e^{m|Z|_{s}} ∥_{L^{p}}^{ρ}ds)^{\frac{1}{ρ}} (C.12)

by 5.2.18:  ≤ C_{p}^{⋄}dLm^{r+1}B′·max_{ρ=1^{⋄},2}(T_{k}/δ)^{1/ρ} · (∫_{0}^{δ} (s^{(r+1)/2} e^{M′s})^{ρ}ds)^{\frac{1}{ρ}}

≤ C_{p}^{⋄}dLm^{r+1}B′(T_{k}∨\sqrt{T_{k}})e^{M′δ}·(\frac{δ^{−1}δ^{\frac{(r+1)}{2} +1}}{(r+1)/2 +1} ∨ \frac{δ^{\frac{−1}{2}} δ^{r/2 +1}}{(r/2 + 1)/2})

≤ C_{p}^{⋄}dLm^{r+1}B′e^{M′δ}δ^{(r+1)/2} × (T_{k} ∨ \sqrt{T_{k}}) .

From this it is evident that there is a constant B^{(2)} = B_{d,m,p,r}^{(2)} such that

∥D_{t}^{(2)⋆}∥_{L^{p}} ≤ B^{(2)} · δ^{(r+1)/2} × (t ∨ \sqrt{t}) ∀ t ≥ 0

for small δ. Multiplying this by e^{−Mt} and taking the supremum over t ≥ 0 gives

D^{(2)} _{p,M}^{⋆} ≤ B^{(2)} · δ^{(r+1)/2} .

for sufficiently small δ > 0 and suitable constant B^{(2)} . Equality (C.12) is justified by the observation that Z_{s} and Z_{T_{k}+s} − Z_{T_{k}} have the same distribution (exercise 3.9.7).

It is time to estimate D^{(1)} . On an interval ((T_{k}, T_{k+1}]] we have

𝔘′[X′]_{t} = ξ′[X′_{T_{k}}, 1; f·(Z_{t}−Z_{t}^{T_{k}}); ξ′]
= 𝔘′[X′]_{T_{k}} + (ξ′[X′_{T_{k}}, 1; f·(Z_{t}−Z_{t}^{T_{k}}); ξ′] − X′_{T_{k}})
and  𝔘[X′′]_{t} = 𝔘[X′′]_{T_{k}} + ∫_{T_{k}}^{t} \overline{f}_{η}(X′′_{s}) dZ_{s}^{η}
= 𝔘[X′′]_{T_{k}} + (ξ[X′_{T_{k}}, 1; f·(Z_{t}−Z_{t}^{T_{k}}); ξ′] − X′_{T_{k}}) .
Thus |D_{t}^{(1)} − D_{T_{k}}^{(1)}| = |ξ′[X′_{T_{k}}, 1; f·(Z_{t}−Z_{t}^{T_{k}}); ξ′] − ξ[X′_{T_{k}}, 1; f·(Z_{t}−Z_{t}^{T_{k}}); ξ′]|
≤ (\underline{m}[f·(Z_{t}−Z_{t}^{T_{k}}); ξ′]·1)^{r+1} × e^{m[f·(Z_{t}−Z_{t}^{T_{k}});ξ′]·1} ,
using (C.10):  ≤ (m · |Z_{t}−Z_{t}^{T_{k}}|)^{r+1} e^{m|Z_{t}−Z_{t}^{T_{k}} |} ,

whence

sup_{T_{k}≤t≤T_{k+1}} |D_{t}^{(1)}−D_{T_{k}}^{(1)}| ≤ (m · |Z−Z^{T_{k}}|_{T_{k+1}}^{⋆})^{r+1} e^{m|Z−Z^{T_{k}} |_{T_{k+1}}^{⋆}}
and D_{T_{k+1}}^{(1)⋆} − D_{T_{k}}^{(1)⋆} ≤ (m · |Z−Z^{T_{k}}|_{T_{k+1}}^{⋆})^{r+1} e^{m|Z−Z^{T_{k}} |_{T_{k+1}}^{⋆}} .
Hence  D_{T_{k}}^{(1)⋆} ≤ ∑_{0≤κ<k}(m · |Z−Z^{T^{κ}}|_{T_{κ+1}}^{⋆})^{r+1} e^{m|Z−Z^{T^{κ}} |_{T_{κ+1}}^{⋆}} .

Now the distribution of |Z−Z^{T^{κ}}|_{T_{κ+1}}^{⋆} is exactly that of Z_{δ}^{⋆}, thus

∥D_{T_{k}}^{(1)⋆}∥_{L^{p}} ≤ m^{r+1}k · ∥|Z|_{δ}^{⋆} e^{m|Z|_{δ}^{⋆}} ∥_{L^{p}}

by 5.2.18:  ≤ m^{r+1}T_{k}δ^{−1} · B′δ^{(r+1)/2}e^{M′δ} .

From this it is evident that there is a constant B^{(1)} = B_{d,m,p,r}^{(1)} such that

for small δ

∥D_{t}^{(1)⋆}∥_{L^{p}} ≤ B^{(1)} · δ^{r/2−1/2} × t ∀ t ≥ 0 .

5.5.5 Let 𝕃^{(n)} be a sequence of laws in 𝔏 =^{def} {(Z., X.)[ℙ] : X. ∈ 𝕏_{AB}}. There is a subsequence, which we will again denote by 𝕃^{(n)} , such that 𝕃^{(n)}(ϕ) ⟶_{n→∞} 𝕃^{(∞)}(ϕ) for all ϕ ∈ C_{b}(𝒞^{d+n}[0, ∞)), defining a positive linear functional 𝕃^{(∞)} of mass one on C_{b}(𝒞^{d+n}[0, ∞)). Due to Kolmogorov’s theorem A.3.17 and the established tightness of the projection of 𝕃^{(∞)} on the 𝒞^{d+n}[0, u], u ∈ ℕ, 𝕃^{(∞)} is σ-additive; due to the polish nature of 𝒞^{d+n}[0, ∞) it is even tight. Thus it belongs to the weak closure of {𝕃^{(n)}} in the set of probabilities on 𝒞^{d+n}[0, ∞). This argument shows that 𝔏 is relatively compact in the topology of weak convergence of measures. The uniform tightness now follows from exercise A.4.8. 5.6.2 (i)

dD\overline{D} = D_{.−}d\overline{D} + dD\overline{D}_{.−} + [D, \overline{D}]

= D_{.−}d\overline{Y} \overline{D}_{.−} + D_{.−}dY \overline{D}_{.−} + D_{.−}d[Y, \overline{Y}]\overline{D}_{.−}

= D_{.−}(−dY + d^{c}[Y, Y] + dJ)\overline{D}_{.−} + D_{.−}dY \overline{D}_{.−}

+ D_{.−}(−d[Y, Y] + d[Y,^{c}[Y, Y]] + d[Y, J])\overline{D}_{.−}

= D_{.−}(d^{c}[Y, Y] + dJ)\overline{D}_{.−} + D_{.−}(−d[Y, Y] + d[Y, J])\overline{D}_{.−}

= D_{.−}(dJ −^{j}[Y, Y] + d[Y, J])\overline{D}_{.−} = D_{.−}(ΔJ − (ΔY)^{2} + ΔYΔJ)\overline{D}_{.−}

= D_{.−}((I + ΔY)ΔJ − (ΔY)^{2})\overline{D}_{.−} = 0 .

(ii) is evident. (5.6.10) Let ℝ^{n} ∋ x_{n} → x. Then X_{t}^{x_{n}} → X_{t}^{x} in probability (use inequality (5.2.36), if necessary after a change of measure that turns Z^{t} into an L^{2}-integrator). Next let ϕ ∈ C_{00}(ℝ^{n}), supported on the ball B_{r}(0) of radius r, say. Then the probability that X.^{x} enters B_{r}(0) before time t is

less than C^{(5.2.25)}|f|_{∞p}e^{Mt}/(|x| − r) ⟶_{x→∞} 0. This implies T_{t}ϕ(x) ⟶_{x→∞} 0.

Approximating an arbitrary ϕ ∈ C_{0} uniformly by functions of compact support will establish the claim T_{t}ϕ ∈ C_{0} . 5.7.2 (ii) Let s < t, say.

𝔼^{x}[|ϕ(X_{t}) − ϕ(X_{s})|^{2}] = 𝔼^{x}[ϕ^{2}(X_{s}) − 2ϕ(X_{s}) · ϕ(X_{t}) + ϕ^{2}(X_{t})]
condition on F_{s} and use (5.7.1): = 𝔼^{x}[(ϕ^{2} − 2ϕ · T_{t−s}(ϕ) + T_{t−s}(ϕ^{2})) ∘ X_{s}]
by equation (5.7.1):  = T_{s}(ϕ^{2} − 2ϕ · T_{t−s}(ϕ) + T_{t−s}(ϕ^{2}))(x) .

This converges to zero uniformly in x as (t − s) → 0 on the grounds that the argument of T_{s} converges uniformly to 0 and the operator norm ∥T_{s} ∥ is less than 1. For the second claim observe that

T_{u−τ} ϕ ∘ X_{τ} − (T_{u−t}ϕ ∘ X_{t} = T_{u−τ} ϕ ∘ X_{τ} − T_{u−t}ϕ ∘ X_{τ}
+ T_{u−t}ϕ ∘ X_{τ} − T_{u−t}ϕ ∘ X_{t}

can be made arbitrarily small in L^{2}(ℙ^{x}) if τ is chosen sufficiently close to t: given ϵ > 0 choose t′ ∈ (t, u) so that t < τ < t′ implies that T_{u−τ} ϕ and T_{u−t}ϕ differ uniformly by less that ϵ/2; this will make the L^{2}(ℙ^{x})-mean of the first summand less than ϵ/2; then use the continuity of the curve τ ↦ (T_{u−t}ϕ)(X_{τ}) to make the L^{2}(ℙ^{x})-mean of the second summand less than ϵ/2 as well. (iii) Let 0 ≤ s < t. Since

e^{−αt}T_{t−s}U_{α}γ = e^{−αt}T_{t−s} ∫_{0}^{∞}e^{−ασ}T_{σ}γ dσ = e^{−αt} ∫_{0}^{∞}e^{−ασ}T_{σ+t−s}γ dσ
σ → σ − t + s:  = e^{−αs} ∫_{t−s}^{∞}e^{−ασ}T_{σ}γ dσ ≤ e^{−αs}U_{α}γ ,

equation (5.7.1) gives

𝔼^{x}[Z_{t}|F_{s}] = 𝔼^{x}[e^{−αt}U_{α}γ ∘ X_{t}|F_{s}] = e^{−αt}T_{t−s}U_{α}γ ∘ X_{s}  ℙ^{x}-a.s.
≤ e^{−αs}U_{α}γ ∘ X_{s} = Z_{s} .

5.7.6 There is an increasing sequence ψ_{n} ∈ C_{00}(E) with pointwise limit |ψ|. Since 𝔼^{x}[ψ_{n} ∘ X_{t}] = T_{t}ψ_{n}(x) ≤ T̆_{t}|ψ|(x) < ∞, the random variables ψ(X_{t}) etc. are all ℙ^{x}-integrable. Now

𝔼^{x}[ψ(X_{t}) − ψ(X_{s})|F_{s}] = (T̆_{t−s}ψ − ψ) ∘ X_{s}  ℙ^{x}-a.s.
= ∫_{s}^{t} (T̆_{τ−s}Ăψ) ∘ X_{s} dτ

= ∫_{s}^{t} 𝔼^{x}[Ăψ ∘ X_{τ} |F_{s}] dτ  ℙ^{x}-a.s.

= 𝔼^{x}[∫_{s}^{t} Ăψ ∘ X_{τ} dτ|F_{s}]  ℙ^{x}-a.s. .

5.7.8 A differs ℙ^{x}-negligibly from a set A^{ℙ^{x}} ∈ F_{0}[X.]. F_{0}[X.] is generated by X_{0} , which equals x ℙ^{x}-almost surely, and so equals the σ-algebra of ℙ^{x}-negligible sets and their complements. For the second claim observe that T^{B} is an F_{.+}^{𝔓}[X.]-stopping time (corollary A.5.12); so A =^{def} [T^{B} = 0] belongs

to F_{⟵}_{0}^{𝔓}[X.] = F_{0}^{𝔓}[X.].

5.7.12 (i) There is an increasing sequence of compact sets K_{n} whose interiors exhaust E. The paths ω which are in K_{n} at all times form a compact set K_{n} in the product topology, which is the topology of pointwise convergence. The bounded continuous cylinder functions based on finite sets σ ∈ [0, ∞) form an algebra that separates the points of K_{n}. By theorem A.2.2 there is one of them, say F_{n} = ϕ_{n} ∘ X_{σ_{n}}, with |F − F_{n}| < 1/n uniformly on K_{n}. Set τ = ⋃ σ_{n} ∪ ℚ and regard F_{n} as a continuous bounded cylinder function based on τ: F_{n} = f_{n} ∘ X_{τ}, where f_{n} = ϕ_{n} ∘ π_{σ_{n}}^{τ}. Clearly f =^{def} lim f_{n} exists uniformly on ⋃_{n} X_{τ} (K_{n}) and F = f ∘ X_{τ} on bounded paths. To see that f is continuous on E_{τ}, let x.^{α} → x. pointwise. Pick a bounded path x.^{0} on all of [0, ∞) and define \overline{x}_{t}^{α} to equal x_{t}^{α} for t ∈ τ, x_{t}^{0} otherwise, and similary define \overline{x}. Then \overline{x}.^{α} → \overline{x}. , and therefore f(x^{α}) = F (\overline{x}^{α}) → F (\overline{x}) = f(x).

(ii) Since τ is dense in [0, ∞) and the paths of 𝒟_{b}(E) are right-continuous, the cylinder functions Φ based on finite subsets of τ (and restricted to 𝒟_{b}(E)) form an algebra A that generates F_{∞}. Since A is countably generated ℙ^{x} is order-continuous on it. Since x ↦ 𝔼^{x}[Φ] is continuous for Φ ∈ A and F is continuous in the topology generated by A on 𝒟_{b}(E), x ↦ 𝔼^{x}[F] is continuous as well (proposition A.4.1). The argument for (iii) is similar. 5.7.13 Let K_{n} be an increasing sequence of compacta whose interiors exhaust E, and set Ω_{n} = {ω : ω_{q} ∈ K_{n} for ℚ ∋ q ≤ t + 1}. Since 𝒟_{b}(E) = ⋃ Ω_{n}, ℙ^{x}[Ω_{n}] > 1 − ϵ for sufficiently large n. The right-continuity of ω causes ω_{s} ∈ K_{n} ∀ s ≤ t. A.2.3 (i) The condition is clearly sufficient. For the necessity suppose then that the sequence (ϕ_{n}) in E or \overline{E} converges uniformly on A to f. By taking a subsequence we may assume that |f − ϕ_{n} | ≤ 2^{−n−1} on A. Then

|ϕ_{n+1} − ϕ_{n} | ≤ 2^{−n} on A, and the function \overline{ψ}_{n} =^{def} −2^{−n} ∨(ϕ_{n+1} −ϕ_{n})∧2^{−n},

which by theorem A.2.2 (i) belongs to \overline{E}, equals ϕ_{n+1} − ϕ_{n} on A_{0} . Therefore the sum \overline{ϕ} =^{def} ϕ_{1} + ∑_{n=1}^{∞} \overline{ψ}_{n} converges uniformly, with limit in \overline{\overline{E}} = \overline{E}. Clearly f = \overline{ϕ} on A_{0} . (ii) Given ϵ > 0 find ϕ_{1}, ϕ_{2} ∈ E with \overline{ρ}(f_{i}, ϕ_{i}) < ϵ on A, i = 1, 2. Then set M =^{def} sup(|ϕ_{1}| ∨ |ϕ_{2}|, and on [−M, M] × [−M, M] approximate (x, y) ↦ \overline{ρ}(x, y) uniformly to within ϵ by a polynomial p(x, y). Then |\overline{ρ}(f_{1}, f_{2}) − p(ϕ_{1}, ϕ_{2})|≤ |\overline{ρ}(f_{1}, f_{2}) − \overline{ρ}(ϕ_{1}, ϕ_{2})|+|\overline{ρ}(ϕ_{1}, ϕ_{2}) − p(ϕ_{1}, ϕ_{2})|≤ 3ϵ on A. A.2.6 Ê consists of functions of compact support, and ˆ \overline{ϕ} is a function of compact support K as well. There exists ψ̂ ∈ Ê=C_{00}(B̂) with ψ̂ ≥ K. Tietze’s extension theorem provides a function ρ̂ ∈ \overline{E} equal to 1/ψ̂ on K. If Ê ∋ ρ̂_{n} → ρ̂ and Ê ∋ ϕ̂_{n} → ˆ \overline{ϕ} uniformly, then the (ϕ̂_{n} · ψ̂ρ̂_{n}) ∘ j ∈ E form a E-confined sequence converging uniformly to \overline{ϕ}. A.2.9 We check the σ-continuity at 0 (see exercise 3.1.5 on page 90). ⇐: Let E ∋ X_{n} ↓ 0. Then k̂ =^{def} inf X̂_{n} vanishes on j(B) and is the pointwise infimum of a sequence in Ê. Consequently θ(X_{n}) = θ̂(X̂_{n}) → ∫ k̂ dθ̂ = 0: θ is indeed σ-additive. ⇒: If k̂ : B̂ → ℝ vanishes on j(B) and is the pointwise infimum of a sequence (X̂_{n}) in Ê then X_{n} =^{def} X̂_{n} ∘ j ↓ 0 on B and so ∫ k̂ dÎ = lim Î(X̂_{n}) = lim I(X_{n}) = 0. A.2.10 The assumption of weak σ-additivity means that for all continuous linear functionals g : L^{p} → ℝ, θ =^{def} g ∘ I is σ-additive. The extension theory of Sections 3.1–3.2 furnishes an integral ∫ . dÎ of the σ-additive Gelfand transform Î (see assumption 3.1.4). Let then E ∋ X_{n} ↓ 0, and set k̂ =^{def} inf X̂_{n}. Then f =^{def} lim I(X_{n}) = lim Î(X̂_{n}) = ∫ k̂ dÎ exists in the norm topology L^{p}, due to the Dominated Convergence Theorem. Since ⟨g|f⟩ = lim⟨g|I(X_{n})⟩ = lim θ(X_{n}) = 0 for all g in the dual of L^{p}, f = 0 thanks to the Hahn–Banach theorem A.2.25. A.2.16 part (iii): In view of part (i) we may as well assume that E contains the constants and is uniformly closed. Let E_{0} ,

Π = ∏_{ψ∈E_{0}} [−∥ψ ∥_{u}, +∥ψ ∥_{u} ] ,

j : E → Π, and \overline{E} = \overline{j(E)} be as in the proof of theorem A.2.2 on page 369. Π and \overline{E} are compact. Let us denote by 𝔲 the E-uniformity on E and by \overline{𝔲} the unique uniformity of the compact space \overline{E}, the C(\overline{E})-uniformity.

(a) Observe first that \overline{ϕ} ↦ \overline{ϕ} ∘ j is an isometric (for the sup–norm) algebra isomorphism of C(\overline{E}) with E. (If j is injective this says that E consists exactly of the restrictions to E of the continuous functions on \overline{E}.)

(b) For a real valued function ϕ denote by d_{ϕ} the pseudometric

(x, y) ↦ d_{ϕ}(x, y) =^{def} |ϕ(x) − ϕ(y)| .

Consider the bases 𝔲_{0} =^{def} \{d_{ϕ} : ϕ ∈ E\} and \overline{𝔲}_{0} =^{def} d\overline{ϕ} : \overline{ϕ} ∈ C(\overline{E})\} for the uniformities 𝔲 and \overline{𝔲}, respectively. They are in one–to–one correspondence via

d_{ϕ} = d\overline{ϕ} ∘ j×j : (x, y) ↦ d\overline{ϕ}(j(x), j(y)) ,  ϕ = \overline{ϕ} ∘ j .

It is easy to see that, for any \overline{d} in the saturation \overline{𝔲} of \overline{𝔲}_{0} , \overline{d} ∘ j×j belongs to 𝔲. Conversely, for d ∈ 𝔲 set

\overline{d}(ξ, η) =^{def} lim d(j^{−1}(𝔙_{ξ}), j^{−1}(𝔙_{η})) ,  ξ, η ∈ \overline{E} ,

where 𝔙_{ξ} denotes the neighborhood filter of ξ, considered as usual as a family of idempotent functions (convention A.1.5 on page 364). Since j(E) is dense in \overline{E}, j^{−1}(𝔙_{ξ}) and j^{−1}(𝔙_{η}) are filters, in fact Cauchy filters on E, so the limit \overline{d}(ξ, η) exists. \overline{d} is a pseudometric and belongs to \overline{𝔲}. Indeed, let ϵ > 0. There are d_{1}, …, d_{n} ∈ 𝔲_{0} and δ > 0 so that max_{i}d_{i}(x, y) < δ ⟹ d(x, y) < ϵ. The d_{i} are of the form d_{i} = \overline{d}_{i} ∘ j×j with \overline{d}_{i} ∈ \overline{𝔲}_{0} ; then clearly max_{i}\overline{d}_{i}(ξ, η) < δ ⟹ \overline{d}(ξ, η) ≤ ϵ. Now d = \overline{d} ∘ j×j:

𝔲 = \overline{𝔲} ∘ j×j .

From this it is obvious that j : E → \overline{E} is uniformly continuous. (If j is injective this says that 𝔲 is the uniformity induced on E from the uniformity of \overline{E}.

(c) Being compact, \overline{E} is complete: a Cauchy filter on \overline{E} is contained in some ultrafilter, which converges; its limit is the limit of the given Cauchy filter.

(d) To see that the compact Hausdorff space \overline{E} with its unique uniformity is the completion of (E, 𝔲_{E}), let f : E → Y be some uniformly continuous map into a complete uniform space (Y, 𝔳). We define the extension \overline{f} : \overline{E} → Y as follows: For ξ ∈ \overline{E}, the neighborhood filter \overline{𝔙_{ξ}} is Cauchy and j^{−1}(\overline{𝔙_{ξ}}) is a Cauchy filter on E. Therefore its forward image f(𝔙) is a Cauchy filter in Y and has a limit, and that shall be \overline{f}(ξ). In other words,

\overline{f}(ξ) =^{def} lim f(j^{−1}(𝔙_{ξ}) .

Clearly \overline{f} ∘ j = f. It is left to be shown that \overline{f} : \overline{E} → Y is uniformly continuous. Let then d′ ∈ 𝔳 and ϵ > 0. There are a δ > 0 and a d = \overline{d} ∘ j×j ∈ 𝔲 such that d(x, y) < 3δ implies d′(f(x), f(y)) < ϵ/3. Let then ξ, η be any two points in \overline{E} with \overline{d}(ξ, η) < δ. There are points x, y ∈ E with j(x), j(y) so close to ξ, η, respectively, that \overline{d}(ξ, j(x)) < δ, \overline{d}(η, j(y)) < δ, d′(\overline{f}(ξ), f(x)) < ϵ/3, and d′(\overline{f}(η), f(y)) < ϵ/3. Then d(x, y) < 3δ and therefore

d′(\overline{f}(ξ), \overline{f}(η)) ≤ d′(\overline{f}(ξ), f(x)) + d′(f(x), f(y)) + d′(f(y), \overline{f}(η))
≤ ϵ/3 + ϵ/3 + ϵ/3 = ϵ :

\overline{f} is uniformly continuous. This establishes that j : (E, 𝔲) → (\overline{E}, \overline{𝔲}) has the universality property of a Hausdorff completion. A.2.19 (iv) implies (ii), for lower semicontinuity: Let x ∈ E, set r = f(x), and let ϵ > 0. The function that has on [f ≤ r − ϵ] the value inf f and at x the value r − ϵ is continuous on the closed set [f ≤ r − ϵ] ∪ {x}. Tietze’s extension theorem provides a continuous extension ϕ to all of E with values in [inf f, r − ϵ]. Clearly ϕ ≤ f and ϕ(x) ≥ f(x) − ϵ. A.2.24 Let U[P] be the algebra of lemma A.2.20, and j : P → P̂ as furnished by theorem A.2.2 (ii). Since 1 ∈ U[P], P̂ is compact; since U[P] is countably generated, P̂ is metrizable; since U[P] generates the topology of P, j is a homeomorphism of P with its image j(P). By exercise A.2.23, j(P) is a G_{δ}-set of P̂; and in the compact metric space P̂ every open set is a K_{σ}-set. A.2.25 (i) We start with the case that A is a linear subspace and B is open. (In this case the result is known as the theorem of Mazur.) Zorn’s lemma provides a maximal linear subspace M containing A and disjoint from B. M is evidently closed. We have to show that the quotient Ṽ =^{def} V/M is one– dimensional and therefore linearly homeomorphic with ℝ. Let p : V → Ṽ be the quotient map and assume by way of contradiction that Ṽ has dimension > 1. Ṽ contains no linear subspace L disjoint from the open convex set B̃ =^{def} p(B) other than {0}; else p^{−1}(L) would be a linear subspace of V disjoint from B and properly containing M. Now the punctured space Ṽ_{∗} =^{def} Ṽ \ {0} is connected, since its intersection with every two–dimensional subspace of Ṽ is homeomorphic with ℝ_{∗}^{2} , and it contains the open convex cone C̃ =^{def} ⋃_{λ>0} λB̃, which is therefore not closed in Ṽ_{∗}; there exists a boundary point x of C̃ in Ṽ_{∗}. No positive scalar multiple λx, λ ≥ 0 belongs to C̃, and neither does −λx: if it did then −x ∈ C̃ and the whole segment [−x, x) =^{def} {λx : λ ∈ [−1, 1)} would lie in C̃, in particular then 0 = 0x ∈ C̃, which is false. That is to say, the whole closed linear subspace ℝx =^{def} {λx : λ ∈ ℝ} is disjoint from B̃, and there is a contradiction: we must indeed have dim Ṽ = 1. We compose p with a suitable linear homeomorphism Ṽ → ℝ to arrive at a continuous linear functional f : V → ℝ which vanishes on A and has f(B) ⊂ (0, ∞), then pick c = 0. The pair (f, c) answers the description. In the case that A is an arbitrary closed convex set and B is still open, we apply Mazur’s theorem with B replaced by the open convex set B−A =^{def} {b− a : b ∈ B , a ∈ A} and with A = {0}. The continuous functional f found above has f(b) > f(a) for all b ∈ B and a ∈ A. Taking c =^{def} inf{f(b) : b ∈ B} yields the claim. In the case that A is an arbitrary closed convex set and B is compact, consider the open sets A + V =^{def} {a + x : a ∈ A , x ∈ V }, where V runs through the convex open symmetric (V = −V ) neighborhoods of zero. Their intersection A with B is void, so one of them, say A+V , has void intersection with B. Apply the previous result to A and the open convex set B − V . (ii) Let f be a linear functional defined and continuous on the linear subspace M ⊂ V. Let A be the closure of f^{−1}({0}) and B =^{def} {b}, where b ∈ M has f(b) = 1 — such b exists and lies outside A except in the trivial case that f = 0. Now (i) provides a continuous linear functional f′ : V → ℝ and c a scalar so that f′(a) ≤ c for a ∈ A and f′(b) > c. Since A is a subspace, f′ = 0 on A, which implies c ≥ 0, and we may assume c = 0. Then F =^{def} f′/f′(b) is the desired extension of f to all of V. (iii) Suppose A is convex and closed in V. For every b ∉ A there are, by (i), a continuous linear functional f_{b} and a constant c_{b} so that A ⊆ [f_{b} ≤ c_{b}]. In other words, A is the intersection of the weakly closed halfspaces [f_{b} ≤ c_{b}], b ∉ A, and is thus weakly closed itself. (iv) A set F of linear functionals on V is equicontinuous if ⋂_{f∈F} [|f| ≤ 1] is a neighborhood of zero in V. For instance, if V is a seminormed space with seminorm ∥ ∥_{V}, then F is equicontinuous if and only if it is uniformly bounded on the unit ball of V: s =^{def} sup{|f(x)| : f ∈ F , ∥x∥_{V} ≤ 1}< ∞; then the ∥ ∥_{V}-ball of radius s^{−1} is contained in ⋂_{f∈F} [|f| ≤ 1]. In partivular the unit ball {x^{∗} ∈ V^{∗} : ∥x^{∗} ∥_{V^{∗}} ≤ 1} is equicontinuous (here ∥x^{∗}∥_{V^{∗}}

=^{def} sup |x^{∗}(x)| : ∥x∥_{V} ≤ 1}.)

Let then F ⊂ V be equicontinuous and set V =^{def} ⋂_{f∈F} [|f| ≤ 1]. Then every x ∈ V is absorbed by V , say x ∈ λ_{x}V , and therefore f(x) ∈ I_{x} =^{def} [−λ_{x}, λ_{x}].

Let now U be an ultrafilter on F. Then the sets U(x) =^{def} {f(x) : f ∈ F}, U ∈ U, form an ultrafilter in the compact interval I_{x} that has a limit g(x) (see exercise A.2.13 on page 374). It is easy to see that x ↦ g(x) is linear and that g(V) ⊆ [−1, 1], so that g is continuous. That is to say, g is in the weak^{∗}–closure of F (see item A.2.32 on page 381), showing that that closure is weak^{∗}–compact. As a corollary, the unit ball of a reflexive Banach space E, being the unit ball of the dual of E^{∗}, is weakly compact. A.2.29 Set Φ_{0}(r) = sup\begin{cases}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{cases}′ : \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned} < r\}. This is clearly an increasing positive numerical function on ℝ_{+} satisfying

\begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}′ ≤ Φ_{0}(\begin{matrix}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{matrix}) ,  f ∈ V .

Given an ϵ > 0, we can find a \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}−δ-ball \{f : \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned} < δ\} contained in the \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}′−ϵ-ball \{f : \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}′ < ϵ\}. Clearly r < δ implies Φ_{0}(r) < ϵ: Φ_{0} has the desired property Φ(r) ⟶_{r→0} 0. Now Φ_{0} may not be right-continuous, but

Φ(r) =^{def} inf(Φ_{0}(s) : s > r)

is, and it retains the other properties of Φ_{0} . A.2.36 The right-continuous version of x. will satisfy the same inequality, so we may as well assume x. to be right-continuous to start with. Set α =^{def} 2(A/B +C) and β =^{def} max_{ρ=p,q}(2B)^{ρ}/ρ. Then ξ_{λ} =^{def} A/B +x_{λ} satisfies

ξ_{μ} ≤ \frac{α}{2} + max_{ρ=p,q} B(∫_{0}^{μ} ξ_{λ}^{ρ} dλ)^{1/ρ} ,  μ ≥ 0 ,

and the claim follows from ξ_{λ} ≤ αe^{βλ} ∀ λ. This is certainly true in some neighborhood of λ = 0. If Λ =^{def} {inf λ : ξ_{λ} > αe^{βλ}} < ∞, then by rightcontinuity

αe^{βΛ} ≤ ξ_{Λ} ≤ \frac{α}{2} + max_{ρ=p,q} αB(∫_{0}^{Λ} e^{βρλ} dλ)^{1/ρ}

< \frac{α}{2} + max_{ρ=p,q} \frac{αBe^{βΛ}}{(βρ)^{1/ρ}} ≤ \frac{α}{2} + \frac{α}{2}e^{βΛ} ≤ αe^{βΛ} .

This contradiction shows that Λ = ∞. A.2.40 Cover U by bounded sets. A.2.50 Chain Rule: If Φ : D → E is differentiable and F : E → S weakly differentiable, then F ∘ Φ : D → S is weakly differentiable and DF ∘ Φ[u] = DF [Φ(u)] DΦ[u]. For c) apply the chain rule and the mean value theorem to t ↦ F (u + t(v − u)). A.3.1 (i): Let d be a metric defining the topology. A closed set F is the zeroset of the function x ↦ d(x, F) =^{def} inf{d(x, y) : y ∈ F }, which is continuous. Therefore F is a Baire set. Then so is every open set. (ii): First the case that G is metrizable. Let U ⊂ G be open and set U_{k} =^{def} {x ∈ G : d(x, U^{c}) ≥ 1/k} ⊆ U. The interiors of these closed sets exhaust U. Therefore f^{−1}(U) = ⋃_{k} ⋃_{N} ⋂_{n>N} f_{n}^{−1}(U_{k}) ∈ F – since {B : f^{−1}(B) ∈ F} is a σ-algebra containing the open sets, it contains the Borels. In the general case consider Γ(B) ∈ F}. This is a σ-algebra. If ϕ ∘ f is F-measurable for all ϕ ∈ C_{ℝ}(G) then Γ contains the sets {ϕ^{−1}(B_{0})} for any ϕ ∈ C_{ℝ}(G) and any B_{0} ∈ B^{∗}(G), if ϕ ∘ f is F/B^{∗}(ℝ)-measurable for all ϕ ∈ C_{ℝ}(G). and thus contains G. That is to say, f : F → G is F/B^{∗}(G)-measurable iff ϕ∘f is F-measurable for all ϕ ∈ C_{ℝ}(G). So if the f_{n} are F/B^{∗}(G)-measurable and converge pointwise to f, then ϕ∘f = lim ϕ∘f_{n} is measurable for all ϕ ∈ C_{ℝ}(G), and so f is F/B^{∗}(G)-measurable. (iii): This counterexample is from [27], page 96, and was pointed out to me by Oliver Diaz–Espinoza. Let f = I be the unit interval, and equip G = I^{I} with the topology of pointwise convergence. For every x ∈ I let f_{n}(x) ∈ I^{I} be the function y ↦ max(0, 1 − n|x − y|). The maps f_{n} : I → I^{I} are continuous, but their pointwise limit f, which maps every x ∈ I to 1_{{x}} : y ↦ [x = y] is not Borel measurable. A.3.2 (i) The functions f of this description form a sequentially closed family. (ii) Suppose sup_{n} f_{n} > 0. Then f^{(n)} =^{def} 1 ∧ n sup_{ν<n} |f_{ν}| ∈ E and 1 = lim f^{(n)} ∈ E^{σ}. Conversely, if 1 ∈ E^{σ}, then there is a countable family {f_{1}, f_{2}, …} ⊂ E whose sequential closure contains 1. The f_{n} cannot all vanish at any one point because then so would 1, thus sup |f_{n}| > 0. (iii) The E_{α} above are the same whether the sequences occurring in their definition are considered as ℝ-valued sequences that converge pointwise in ℝ or as \overline{ℝ}-valued sequences that happen to have a real-valued pointwise limit. A.3.6 The sequential closure of the class of differences of bounded lower semicontinuous functions contains the topology^{13}, which forms a multiplicative class; it therefore contains all Borel functions. Conversely, a lower semicontinuous function h is the supremum of the countable collection q ·[h > q], q ∈ ℚ, and so is Borel measurable. Then so is a bounded lower semicontinuous function, a difference of such, and every function in the sequential closure of such differences. A.3.7 Suppose f ∈ E^{σ} is E-confined. There is a ψ ∈ E with |f| ≤ ψ. The collection of functions g ∈ E^{σ} such that −ψ ∨ g ∧ ψ is the limit of a bounded E-confined sequence in E^{σ} is sequentially closed and thus contains E^{σ}. A.3.8 (i) This is just the definition of inner measure, which agrees with μ on A^{σ}. (ii) Any idempotent member (set) of the class inf {ḟ ∈ L^{0}(A^{σ}, μ) : ∃f ∈ ḟ with f ≥ Ω′} will do. A.3.10 Due to lemma A.2.20 any decreasingly directed collection Φ ⊂ C_{b}(E) contains a decreasing sequence (ϕ_{n}) with the same pointwise infimum; then inf μ(Φ) = inf μ(ϕ_{n}). For if μ(ϕ) < a for some ϕ ∈ Φ, then ϕ ∧ ϕ_{n} ↓ ϕ and consequently inf μ(ϕ_{n}) ≤ lim μ(ϕ ∧ ϕ_{n}) < a. A.3.21 See [5, page 286 ff.].

A.3.25 There is a collection L of affine functions ℝ_{+} ∋ x ↦ ℓ(x) = ax + b whose pointwise infimum is ϕ. For each one of them b is positive and ∫ ϕ(|z|) dμ ≤ ∫ ℓ(|z|) dμ = a ∫ |z| dμ + b ∫ 1 dμ ≤ a ∫ |z| dμ + b = ℓ( ∫ |z| dμ). Taking the infimum over ℓ ∈ L yields the claim: ∫ ϕ(|z|) dμ ≤ ϕ( ∫ |z| dμ). A.3.26 This is evident if Φ(x, ω) is a product of the form ϕ(x)ψ(ω), then if it is the linear combination of such functions, then if it belongs to the sequential closure of the algebra formed by the latter. A.3.28 Let f^{∗} be an element in the unit ball E_{1}^{∗} of the dual of E. Then

⟨∫ f dν|f^{∗}⟩ = ∫ ⟨f|f^{∗}⟩ dν ≤ ∫ ∥f ∥_{E} d ν .

Taking the supremum over f ∈ E_{1}^{∗} yields the claim. A.3.29 The cases q = ∞ and p = q are trivial. The case 1 = p < q: If ∥∥f ∥_{L^{1}(μ)} ∥_{L^{q}(ν)} > 1, then

∫ (∫ |f(x, y)| μ(dx))^{q} ν(dy) > 1

and there is a function g ∈ L_{+}^{q′}(ν) of norm one in that space with

1 < ∫ (∫ |f(x, y)| μ(dx)) · g(y) ν(dy)
= ∫ (∫ |f(x, y)| · g(y) ν(dy)) μ(dx)

≤ ∫ (∫ |f(x, y)|^{q} ν(dy))^{1/q} · (∫ |g(y)|^{q′} ν(dy))^{1/q′} μ(dx)
= ∥∥f∥_{L^{q}(ν)}∥_{L^{1}(μ)} .

In the remaining case 0 < p < q < ∞ write

\begin{aligned}∥∥f∥_{L^{p}(μ)}∥_{L^{q}(ν)} & = ∥∥|f|^{p}∥_{L^{1}(μ)}^{1/p}∥_{L^{q}(ν)} = ∥∥|f|^{p}∥_{L^{1}(μ)}∥_{L^{q/p}(ν)}^{1/p} \\ & ≤ ∥∥|f|^{p}∥_{L^{q/p}(ν)}∥_{L^{1}(μ)}^{1/p} = ∥∥f∥_{L^{q}(ν)}∥_{L^{p}(μ)} . \end{aligned}

A.3.34 Only the sufficiency may not be obvious. Assume then that (e^{i⟨ξ|x_{n}⟩}) converges for almost all ξ. Then e^{i⟨ξ|x_{n}−x_{m}⟩} ⟶_{m,n→∞} 1.

With  γ(ξ) = e^{−|ξ|^{2}/2}/\sqrt{2π} ,
Q_{m,n} =^{def} ∫ e^{i⟨ξ|x_{n}−x_{m}⟩}γ(ξ) dξ ⟶_{m,n→∞} 1 .
Now  Q_{m,n} = e^{−|x_{n}−x_{m}|/2} ∫_{ℝ^{d}} e^{−|ξ−i(x_{n}−x_{m})|^{2}/2}/\sqrt{2π} dξ

= e^{−|x_{n}−x_{m}|/2} ∫_{ℝ^{d}} e^{−|ξ|^{2}/2}/\sqrt{2π} dξ
= e^{−|x_{n}−x_{m}|/2} .

From the resulting e^{−|x_{n}−x_{m}|/2} ⟶_{m,n→∞} 1 we conclude that (x_{n}) is Cauchy. A.3.37 (i) The continuity of h_{1}⋆μ_{2} is an immediate consequence of the metrizability of G and the Dominated Convergence Theorem. To see that h_{1}⋆μ_{2} vanishes at ∞, let ϵ > 0 be given. There exist a compact set K_{2} so that μ_{2}(K_{2}^{c}) < ϵ and a compact set K_{1} outside which |h_{1}| < ϵ. If g lies outside the compact algebraic sum K_{1} + K_{2} ⊂ G, then h_{1}(g − g_{2}) < ϵ for g_{2} ∈ K_{1} and therefore ∫_{K_{2}} h_{1}(g − g_{2}) μ_{2}(dg_{2}) < ϵ∥μ_{2}∥. Clearly ∫_{K_{2}^{c}} h_{1}(g −

g_{2}) μ_{2}(dg_{2}) < ϵ∥h_{1}∥. Thus h_{1}⋆μ_{2} ⟶_{g→∞} 0.

A.3.44 Since μ is order-continuous (exercise A.3.10), it makes sense to talk about the support C of μ (exercise A.3.13). Let S be a lifting, Υ a countable uniformly dense subset of U[E] (see lemma A.2.20), and N the negligible set ⋃{[Sυ ≠ υ] ∩ C : υ ∈ Υ}. For x ∉ N set T f(x) = Sf(x), f ∈ L^{∞}. If x ∈ N, consider the ideal I_{x} of functions f ∈ L^{∞} that differ negligibly from a function υ′ that is continuous (in the metric topology) at x and has υ′(x) = 0. As in the proof of lemma A.3.40 one checks that there is an x̂ ∈ Ê at which all the functions of Î_{x} vanish. Set T f(x) = f̂(x̂) and check that T is a lifting with T υ(x) = υ(x) for x ∈ C and υ ∈ U[E]. For x ∈ C and ϕ ∈ C_{b} we have by lemma A.2.20 T ϕ(x)≥sup{T υ(x) : ϕ ≥ υ ∈ U[E]}= sup{υ(x) : ϕ ≥ υ ∈ U[E]}= ϕ(x). Applying this to −ϕ gives T ϕ(x) = ϕ(x). A.3.45 Equation (A.3.17): Integrating e^{−(x^{2}+y^{2})/2} over the plane in polar

coordinates establishes first that

∫_{−∞}^{+∞} e^{−x^{2}/2} = \sqrt{2π} .

The variable substitution u = (x − iξt)/\sqrt{t} turns the integral

𝔼[e^{iξX} ] = \frac{1}{\sqrt{2πt}} ∫_{−∞}^{+∞} e^{iξx} · e^{−x^{2}/2t} dx

into

\frac{e^{−tξ^{2}/2}}{\sqrt{2π}} ∫_{−∞}^{+∞} e^{−u^{2}/2} du = e^{−tξ^{2}/2} .

A.3.49 Apply functional calculus to the self-adjoint operator B. Or apply the power series for ∥B ∥^{1/2}\sqrt{1 − x} to the matrix I − B/∥B ∥, ∥B ∥ being the operator norm of B. A.4.8 See [12], ch. IX, §5: Treat first the case that E is also locally compact. Suppose the locally compact space E has a countable basis, and 𝔓 is a family of probabilities on E that is relatively compact in the topology σ(𝔓.(E), C_{00}(E)) of convergence on continuous functions of compact support. There exists an increasing sequence of compacta K_{n} ⊂ E whose interiors cover E. By way of contradiction assume 𝔓 is not uniformly tight. Then there exist an α > 0 and measures μ_{n} ∈ 𝔓 with μ_{n}(K_{n}) ≤ 1 − α. Extracting a subsequence we may assume that μ_{n} → μ ∈ 𝔓.(E) on all ϕ ∈ C_{00}(E). Now μ(K) > 1 − α for some compact K ⊂ E, since μ is a tight probability. There exist a ϕ ∈ C_{00}(E) with ϕ = 1 on K and an N ∈ ℕ so that K̊_{N} contains the support of ϕ. Now as μ_{n}(ϕ) → μ(ϕ) > 1 − α we have μ_{n}(K_{n}) ≥ μ_{n}(K_{N}) ≥ μ(ϕ) > 1 − α for sufficiently large n ≥ N, a contradiction.

An arbitrary polish space is the intersection of a decreasing sequence of open subsets E_{n} of some compact space (exercise A.6.1). We view the measures of 𝔓 as measures on the E_{n}, which are locally compact. Due to the first part of the argument there are compacta K_{n} ⊂ E_{n} with μ(E_{n} − K_{n}) < α2^{−n} ∀ μ ∈ 𝔓. K =^{def} ⋂_{n} K_{n} ⊂ E is compact and has μ(E − K) ≤ α. A.5.6 (ii) Suppose that K′ ⊂ K^{∪_{f}} has the property that no finite subcollection has void intersection. Then there is an ultrafilter U containing K′. Every set K′ ∈ K′ is the finite union K′ = ⋃_{i=1}^{I(K′)} K′_{i} of sets in K. At least one of the K′_{i}, say K′_{i(K′)} , belongs to U. No intersection of finitely many such K′_{i(K′)} is void, and therefore neither is ⋂ {K′_{i(K′)} : K′ ∈ K′} ⊂ ⋂ K′.

(iii) Take for the closed sets the sets in K^{∪_{f} ∩_{a}}. A.5.16 Apply theorem 2.4.7. A.5.18 Proposition 3.5.2 shows that O contains P. Let X ∈ 𝔇. Let δ > 0 and define the stopping times S_{0} = 0 and

S_{k+1} = inf {t > S_{k} : |X − X_{S_{k}}|_{t}^{⋆} ≥ δ } ,  k = 1, 2, …
and  X^{(δ)} = X_{0}[[0]] + ∑_{k} X_{S_{k}} · [[S_{k}, S_{k+1})) .  (∗)

Since X has no oscillatory discontinuities, S_{k} ↑ ∞. Evidently, X and X^{(δ)} differ uniformly by less than δ. The process X^{(δ)} differs from the previsible process

X_{0}[[0]] + ∑_{k} X_{S_{k}} · ((S_{k}, S_{k+1}]]

only in the set ⋃_{k}[[S_{k}]]. The processes (∗) form therefore a generator of O for which the claim is true. It is now easy to check that the optional processes for which the second claim holds is a monotone class. An application of theorem A.3.4 finishes the proof. A.5.20 The family of processes that have an optional projection is a monotone class. It contains the processes of the form [t, ∞)×g, g ∈ F_{∞}^{∗} bounded, which generate the measurable σ-algebra. Indeed, let M^{g} be the rightcontinuous martingale 𝔼[g|F_{t+}^{ℙ}] (proposition 2.5.13). It follows from Doob’s optional stopping theorem 2.5.22 that [[t, ∞)) · M^{g} is an optional projection of [t, ∞) × g. If X^{O,ℙ} and \overline{X}^{O,ℙ} are two optional projections of X, then B =^{def} [X^{O,ℙ} ≠ \overline{X}^{O,ℙ}] is an optional set. If it were not ℙ-evanescent, one could find a stopping time T whose graph is contained in B and has non-negligible projection on Ω. This however is clearly impossible. A.5.21 X^{O,ℙ} meets the description. Indeed, for any t and A ∈ F_{t} = F_{t+}^{ℙ}

𝔼[A · X_{t}^{O,ℙ}] = 𝔼[X_{t_{A}}^{O,ℙ}[t_{A} < ∞]] = 𝔼[X_{t_{A}}[t_{A} < ∞]] = 𝔼[A · X_{t}] :

the F_{t}-measurable random variables X_{t}^{O,ℙ} and X_{t} differ negligibly. A.6.1 (i) See exercise A.2.24. (ii) Let p : P → S be a continuous map from the polish space onto the Suslin set S, and C ⊂ S closed. Then p : p^{−1}(C) → C exhibits C as a Suslin set. (iii) Let S_{n} be Suslin subsets of a Hausdorff space and p_{n} : P_{n} → S_{n} continuous surjections. In the product space ∏_{n} P_{n}, which is polish, let P =^{def} {(x_{n}) : p_{n}(x_{n}) = p_{m}(x_{m}) for m ≠ m}. This is a closed and therefore polish subspace of ∏_{n} P_{n}. The continuous map p : P → ⋂_{n} S_{n} exhibits the intersection of the S_{n} as Suslin. For the union, consider the disjoint topological sum P =^{def} ⨄_{n} P_{n}. Its is again polish, and the obvious map p : P → ⋃_{n} S_{n} shows that the union of the S_{n} is polish. (iv) By (i), a closed ball of the given Suslin space S is Suslin. Since S is separable as the continuous image of a separable space and metrizable by assumption, the complement of a closed ball is the countable union of closed balls and is therefore also Suslin. The collection of sets which together with their complements are Suslin is closed under taking complements and by (ii) is closed under countable unions and contains the closed balls. It contains therefore the σ-algebra generated by the closed balls, the Borels. A.8.1 (i) If \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{0} ≤ a, then there exists a decreasing sequence (λ_{n}) with inf λ_{n} ≤ a and ℙ[|f| > λ_{n}] ≤ λ_{m} for 1 ≤ m ≤ n. Then ℙ[|f| > a] ≤ ℙ[|f| > λ] ≤ λ_{m} for all m, and therefore ℙ[|f| > a] ≤ λ ≤ a. The reverse implication is obvious.

(ii), for p = 0: Let a = \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{0} and b = \begin{aligned}⌈ \\ ⌈ & g & ⌉ \\ & & ⌉ \end{aligned}_{0} . Since [|f + g| > a + b] ⊂ [|f| > a] ∪ [|g| > b], we have by (i) ℙ[|f + g| > a + b] ≤ a + b, i.e.,

\begin{aligned}⌈ \\ ⌈ & f + g & ⌉ \\ & & ⌉ \end{aligned}_{0} ≤ \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{0} + \begin{aligned}⌈ \\ ⌈ & g & ⌉ \\ & & ⌉ \end{aligned}_{0} .

(iii), for p = 0: lim_{r→0} \begin{aligned}⌈ \\ ⌈ & rf & ⌉ \\ & & ⌉ \end{aligned}_{p} = 0 clearly implies ℙ[|f| = ∞] = 0.

(iv), for p = 0: The algebraic properties are obvious. Given a \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{0}-Cauchy sequence (f_{n}) extract a subsequence (f_{n_{k}}) with \begin{aligned}⌈ \\ ⌈ & f_{n_{k+1}} − f_{n_{k}} & ⌉ \\ & & ⌉ \end{aligned} ≤ 2^{−k}. Then ∑_{k} |f_{n_{k+1}} − f_{n_{k}}| is a.s. finite, and therefore (f_{n_{k}}) converges a.s. The limit is a \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{0}-mean limit of (f_{n}) by a standard argument. A.8.2 This is clear if p ≥ 1. For 0 < p < 1 Hölder’s inequality (theorem A.8.4 on page 449) with conjugate exponents 1/p, 1/(1 − p) gives a_{1} + … + a_{n} ≤ n^{1−p}(a_{1}^{1/p} + … + a_{n}^{1/p}) for a_{ν} ≥ 0. Apply this to a_{ν} = (∫ |f_{ν}|^{p}) to obtain

∫ |f_{1} + … + f_{n}|^{p} ≤ ∫ |f_{1}|^{p} + … + ∫ |f_{n}|^{p}

≤ n^{1−p}((∫ |f_{1}|^{p})^{1/p} + … + (∫ |f_{n}|^{p})^{1/p})^{p}

and take p^{th} roots. A.8.4 Hölder’s inequality for the special case r = 1 of conjugate exponents is in any textbook on integration. By applying that to the function |fg|^{r}, with conjugate exponents p/r and q/r, the general case follows. For the second claim reduce to the case ∥f ∥_{p} = 1 and take g = f^{p−1} . A.8.5 Let 1/p and 1/q be points in I_{f} and 1/s = θ/p+(1−θ)/q (0 < θ < 1) a point in between. Set a = sθ/p; then 1 − a = (1 − θ)s/q and 0 < a < 1. Apply Hölder’s inequality with conjugate exponents 1/a and 1/(1 − a) to |f|^{p} and |f|^{q}:

∫ |f|^{s} dμ = ∫ |f|^{pa} · |f|^{q(1−a)} ≤ (∫ |f|^{p} dμ)^{a} · (∫ |f|^{q} dμ)^{(1−a)}
= ∥f∥_{L^{p}}^{ap} · ∥f∥_{L^{q}}^{(1−a)q} = ∥f∥_{L^{p}}^{θs} · ∥f∥_{L^{q}}^{(1−θ)s} ,
and so ∥f∥_{L^{s}} ≤ ∥f∥_{L^{p}}^{θ} · ∥f∥_{L^{q}}^{1−θ}
and ln (∥f∥_{L^{s}}) ≤ θ ln (∥f∥_{L^{p}}) + (1 − θ) ln (∥f∥_{L^{q}}) .

A.8.11 The notions of F-measurability and almost sure finiteness coincide, whether ℙ or ℙ′ is the measure: L^{0}(ℙ) and L^{0}(ℙ′) coincide as sets. Suppose f_{n} → 0 in ℙ-measure, and let ϵ > 0. Now

ℙ′[|f_{n}| > ϵ] = ∫ [|f_{n}| > ϵ] · g′ dℙ

= ∫ [|f_{n}| > ϵ][g′ > M] · g′ dℙ + ∫ [|f_{n}| > ϵ][g′ ≤ M] · g′ dℙ

≤ ∫ [g′ > M] · g′ dℙ + M · ℙ[[|f_{n}| > ϵ]] .

Choosing first M so large that the first summand on the right is less than ϵ/2 and then n so large that the second summand also is less than ϵ/2, we see that eventually \begin{aligned}⌈ \\ ⌈ & f_{n} & ⌉ \\ & & ⌉ \end{aligned}_{L^{0}(ℙ′)} ≤ ϵ: f_{n} → 0 in ℙ′-measure. Interchanging the roles of ℙ and ℙ′ and replacing g′ by g =^{def} (g′)^{−1} shows that the converse implication f_{n} → 0 in ℙ′-measure ⟹ f_{n} → 0 in ℙ-measure is also true: the two topologies are, indeed, the same.

A.8.12 Set Φ(r) = sup\begin{cases}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{cases}_{L^{0}(ℙ′)} : \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{L^{0}(ℙ)} ≤ r\}. To see that Φ(r) ⟶_{r→0} 0

let ϵ > 0 be given and denote by g a Radon–Nikodym derivative dℙ′/dℙ. There is a K > 1 so that ℙ′[g > K] < ϵ/2. Set δ =^{def} ϵ/(2K). If \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{L^{0}(ℙ)} < δ, then ℙ[|f| > ϵ] < ϵ/(2K) and so ℙ′[|f| > ϵ]≤ℙ′[g > K] + Kℙ[|f| > ϵ]< ϵ. To get Φ right-continuous replace it by its right-continuous version. A.8.15 (i) 𝔼[|f|^{p}] = ∫_{0}^{∞} t^{p} dℙ[|f| ≤ t] = ∫_{0}^{∞}(T^{λ+})^{p}[T^{λ+} < ∞] dλ by the change-of-variable theorem 2.4.7, where T^{λ+} =^{def} inf{t : ℙ[|f| ≤ t] > λ} = inf{t : ℙ[|f| > t] ≤ 1 − λ} = ∥f ∥_{[1−λ]} and [T^{λ+} < ∞] = [0 ≤ λ < 1]. Hence

𝔼[|f|^{p}] = ∫_{0}^{1} ∥f ∥_{[1−λ]}^{p} dλ = ∫_{0}^{1} ∥f ∥_{[λ]}^{p} dλ. The last claim is done similarly, with t ↦ t^{p} replaced by t ↦ Φ(t).

(ii) If λ < ∥f + g ∥_{[α+β]} , then

α + β ≤ ℙ[|f + g| > λ] ≤ ℙ[|f| + |g| > λ]
≤ ℙ[|f| > ∥f∥_{[α]}] + ℙ[|f| + |g| > λ , |f| ≤ ∥f∥_{[α]}]
≤ α + ℙ[|g| > λ − ∥f∥_{[α]}] .

Thus β ≤ ℙ[|g| > λ − ∥f ∥_{[α]}], i.e., ∥g ∥_{[β]} ≥ λ − ∥f ∥_{[α]}] and λ ≤ ∥f ∥_{[α]} + ∥g ∥_{[β]} . A.8.16 Let ρ > 0. Then

ρ < ∥∥f∥_{[β;τ]}∥_{[α;ℙ]}
⟹  α < ℙ[∥|f|∥_{[β;τ]} > ρ] = ℙ[τ[|f| > ρ] > β]
⟹  αβ < ∫ τ[f(ω, ·) > ρ] ℙ(dω) = ∫ ℙ[f(·, t) > ρ]τ(dt) .

With g(t) =^{def} ℙ[f(·, t) > ρ] we have 0 ≤ g ≤ 1 and

αβ < ∫ g(t) τ(dt)
= ∫ g(t)[g(t) ≤ γ] τ(dt) + ∫ g(t)[g(t) > γ] τ(dt)
≤ γ + τ[g > γ]
so that  αβ − γ ≤ τ[g > γ] = τ[ℙ[|f| > ρ] > γ] ,
i.e.,  αβ − γ ≤ τ[∥f∥_{[γ;ℙ]} > ρ] ,
which reads  ρ ≤ ∥∥f∥_{[γ;ℙ]}∥_{[αβ−γ;τ]} .

A.8.17 (A.8.1): Set λ = ∥g ∥_{[α]} and denote the right-hand side of the first inequality by Λ. Then

ℙ[f > Λ] ≤ ℙ[f > Λ ; g ≤ λ] + ℙ[g > λ] ≤ ℙ[f^{r}/Λ^{r} > 1 ; λ/g ≤ 1] + α

≤ \frac{λ}{Λ^{r}} 𝔼[f^{r}/g] + α = \frac{λ}{E^{r}Λ^{r}} + α = β + α .

(A.8.2): Set λ = ∥g ∥_{[α/2]} and denote the right-hand side of inequality (A.8.2) by Λ. Then

ℙ[fg > Λ] ≤ ℙ[fg > Λ ; g < λ] + ℙ[g ≥ λ] ≤ ℙ[f > Λ/λ; λ/g > 1] + α/2

≤ λ𝔼[[f > Λ/λ] · 1/g] + α/2 ≤ ∥\frac{λf}{Λ} ∥_{L^{r}(ℙ/g)}^{r} + α/2

≤ \frac{λ^{r+1}E^{r}}{Λ^{r}} + α/2 = α/2 + α/2 = α .

A.8.22 p = 0: If ∥f_{n} ∥_{[α]} ≤ a ∀ n, then ℙ[f_{n} > a] ≤ α ∀ n and consequently ℙ[f > a] = sup_{n} ℙ[f_{n} > a] ≤ α, which says that ∥f ∥_{[_{]}}α ≤ a. A.8.27 In the proof of inequality (A.8.6) replace the constant 2 by any A > 1. The argument gives ∥f ∥_{2} ≤ AK_{1} · ∥f ∥_{[(A−1/AK_{1})^{2}]} . Solving (A − 1/AK_{1})^{2} = κ and using the value K_{1} = \sqrt{2} from remark A.8.28 produces the claim. A.8.29 Since p ↦ Γ((p + 1)/2) is convex, there are two points at which Γ((p + 1)/2) equals \sqrt{π}/2. One of them is p = 2; inspection of a table shows that the other is p_{0} ≈ 1.85. To the left of p_{0} equation (A.8.8) gives K_{p} = 2^{1/p − 1/2} . Calculations on a hand-held calculator show that the ratio

(\sqrt{π}/Γ((p + 1)/2))/2

takes its maximum at p_{m} ≈ 1.92175 and that its p_{m}^{th} root there is approximately 1.000366283. A.8.30 By the Central Limit Theorem

\begin{aligned}lim_{n} \frac{1}{n^{p}} ∫_{T} |ϵ_{1} + … + ϵ_{n}|^{p} dτ & = \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞} |x|^{p}e^{−x^{2}/2} dx \\ & = 2^{p/2} \frac{Γ(\frac{p+1}{2})}{\sqrt{π}} . \end{aligned}

Thus

K_{p} ≥ ( \frac{\sqrt{π}}{Γ( \frac{p+1}{2} )} )^{1/p}/\sqrt{2} .

Also, the choice n = 2 and a_{1} = a_{2} = 1 implies K_{p} ≥ 2^{\frac{1}{p} − \frac{1}{2}} . A.8.32 (Suggested by Roger Sewell) Show that

b(p) = \frac{2 sin(pπ/2)Γ(p)}{π}

and then analyze the right hand side or have the computer draw a graph. A.8.35 Let x_{1}, …, x_{n} ∈ E and γ_{1}^{(q)}, …, γ_{n}^{(q)} symmetric q-stable.

Then ∥∥∑_{ν=1}^{n} v ∘ u(x_{ν}) γ_{ν}^{(q)}∥_{G}∥_{L^{p}(dx)} ≤ T_{p,q}(v) · (∑_{ν=1}^{n} ∥u(x_{ν})∥_{F}^{q} )^{1/q}

≤ T_{p,q}(v) · ∥u∥ · (∑_{ν=1}^{n} ∥x_{ν}∥_{E}^{q})^{1/q} .

and ∥∥∑_{ν=1}^{n} v ∘ u(x_{ν}) γ_{ν}^{(q)}∥_{G}∥_{L^{p}(dx)} ≤ ∥v∥ · ∥∥∑_{ν=1}^{n} u(x_{ν}) γ_{ν}^{(q)}∥_{G}∥_{L^{p}(dx)}

≤ ∥v∥ · T_{p,q}(u) · (∑_{ν=1}^{n} ∥x_{ν}∥_{E}^{q})^{1/q} .

A.9.2

\frac{T_{t}ψ − ψ}{t} = \frac{1}{t}(∫_{0}^{s} T_{σ+t}ϕ dσ − ∫_{0}^{s} T_{σ}ϕ dσ)

= \frac{1}{t}(∫_{t}^{s+t} T_{σ}ϕ dσ − ∫_{0}^{s} T_{σ}ϕ dσ)

= \frac{1}{t}(∫_{s}^{s+t} T_{σ}ϕ dσ − ∫_{0}^{t} T_{σ}ϕ dσ) ⟶_{t→0} T_{s}ϕ − ϕ ,

and so ψ ∈ D[A]

and  Aψ = T_{s}ϕ − ϕ
or  T_{s}ϕ − ϕ = A ∫_{0}^{s} T_{σ}ϕ dσ for all ϕ ∈ C .

Since ∫_{0}^{s} T_{σ}ϕ dσ/s ⟶_{s→0} ϕ, D[A] is dense in C_{0}(E). We have used here that T_{s} ∘ Aϕ = A ∘ T_{s}ϕ for ϕ ∈ D[A], which is left for the reader to establish [34, chapter I]. A.9.4 See [54, page 320]. A.9.8 For every s ∈ [0, ∞), x ∈ E, and β < 1 there is a function ϕ_{s,x,β} ∈ C_{0}(E) with 0 ≤ ϕ ≤ 1 and T_{s}ϕ_{s,x,β}(x) > β. Since s ↦ T_{s}ϕ(x) is continuous, we have T_{s}ϕ_{s,x,β}(x) > β on a whole neighborhood U_{s} of s. Any compact interval [0, k] can be covered by finitely many of them; the supremum ϕ_{x,β}^{k} of the corresponding ϕ_{s,x,β} ’s has T_{s}ϕ_{x,β}^{k}(x) > β for all s ∈ [0, k]. For any fixed α > 0 the choice of a sufficiently large k_{α} will result in αU_{α}ϕ_{x,β}^{k_{α}}(x) > β. This inequality will by continuity hold in a whole neighborhood of x. Now let K ⊂ E be compact. Taking a finite supremum of such functions ϕ_{x,β}^{k_{α}} we find a function ϕ_{α,β}^{K} ∈ C_{0}(E) with 0 ≤ ϕ_{α,β}^{K} ≤ 1 such that αU_{α}ϕ_{α,β}^{K} > β on K. Let K_{n} be an increasing sequence of compacta whose interiors cover E, take α_{n} = 1/n, and choose ϕ_{n} ∈ C_{0}(E) such that ϕ_{n} ≤ 1 and ψ_{n} =^{def} α_{n}U_{α_{n}}ϕ_{n} > 1 − 2^{−n} on K_{n}. Now observe that Aψ_{n} = α_{n}ψ_{n} − α_{n}ϕ_{n} → 0. A.9.9 Applying the identity (αI − A)U_{α} = I to the ψ_{n} of A.9.8 (iii) gives

αU_{α}ψ_{n} − U_{α}Aψ_{n} = ψ_{n}
and in the limit  α ∫_{E} U_{α}(x, dy) = 1 .
i.e.,  α ∫_{0}^{∞} e^{−αs}T_{s}(x, E) ds = 1 ,

78 References

which shows that the measures T_{s}(x,.) all have total mass one. A.9.11 (i), ⇐: Urysohn’s lemma provides positive continuous functions ρ_{n} ≤ 1 of compact support so that [ρ_{n} ≠ 0] ⊂ [ρ_{n+1} = 1] and sup_{n} ρ_{n}(x) = 1 at every point x ∈ E. Let ψ_{n} ∈ C be so that |ϕ| ≤ ψ_{n}, and (s, x) ↦ ∫_{E}^{∗} T_{s}(x, dy) ψ_{n}(y) is finite and continuous on [0, n] × K_{n}. Then

∥ϕ − ϕ · ρ_{k} ∥_{˘ n,K_{n}} = ∥ϕ · (1 − ρ_{k})∥_{˘ n,K_{n}} ≤ ∥ψ_{n} − ψ_{n} · ρ_{k} ∥_{˘ n,K_{n}} ⟶_{k→∞} 0 ,

since the continuous functions (s, x) ↦ ∫_{E}^{∗} T_{s}(x, dy) (ψ_{n}−ψ_{n}·ρ_{k})(y) decrease pointwise, and by Dini’s lemma A.2.1 uniformly on [0, n]×K_{n}, to zero. There is therefore a k_{n} so that ϕ_{n} =^{def} ϕ · ρ_{k_{n}} ∈ C_{00}(E) has ∥ϕ − ϕ_{n} ∥_{˘ n,K_{n}} < 2^{−n}.

Clearly \begin{aligned}⌈ \\ ⌈ & ϕ − ϕ_{n} & ⌉ \\ & & ⌉ \end{aligned}˘ ≤ (n + 1)2^{−n} ⟶_{n→∞} 0.

(i), ⇒: Let ϕ_{n} ∈ C_{00}(E) have ∥ϕ − ϕ_{n} ∥_{˘ n,K_{n}} < 2^{−n}, n = 1, 2, …, and set ϕ_{0} = 0. The function ψ =^{def} ∑_{n≥1} |ϕ_{n} − ϕ_{n−1}| serves simultaneously for all t < ∞ and compact K ⊂ E. (ii) Let ϕ ∈ C̆ and 0 ≤ s ≤ u. Then for every t < ∞ and compact K ⊂ E

\begin{matrix}∥ & T̆_{s}ϕ & ∥ \\  \\ ˘ t,K \end{matrix} ≤ sup \{ ∫_{E}^{∗} T_{τ} (x, dy) T̆_{s}|ϕ| (y) : 0 ≤ τ ≤ t, x ∈ K\}
≤ sup { ∫_{E}^{∗} T_{τ+s}(x, dy) |ϕ|(y) : 0 ≤ τ ≤ t, x ∈ K}
≤ sup { ∫_{E}^{∗} T_{τ} (x, dy) |ϕ|(y) : 0 ≤ τ ≤ t + u, x ∈ K}
= ∥ϕ∥_{˘ t+u,K} ,

showing that T̆_{u} : C̆ → C̆ is continuous. Next let ϕ̆ ∈ C̆ and ϕ ∈ C_{00}(E).

Then T̆.ϕ̆ − T.ϕ = T̆.(ϕ̆ − ϕ) has \begin{matrix}∥ & T̆_{s}ϕ̆ − T_{s}ϕ & ∥ \\ \\ ˘ t,K \end{matrix} ≤ ∥ϕ̆ − ϕ∥_{˘ t+u,K} for

0 ≤ s ≤ u. This shows that the curve T̆.ϕ̆ is on bounded intervals [0, u] the uniform limit of continuous curves T.ϕ in C̆ and therefore is continuous itself. A.9.13 See exercise A.3.16 on page 401. A.9.15 It is evident that T_{t}^{⊢} is linear and has operator norm ≤ 1. To see that it maps C_{0}^{⊢} into itself it suffices to check its behavior on functions of the form (τ, x) ↦ ϕ_{1}(τ)ϕ_{2}(x), ϕ_{1} ∈ C_{0}(ℝ_{+}), ϕ_{2} ∈ C_{0}(E), on the grounds that the linear combinations of these form an algebra A_{0} uniformly dense in C_{0}^{⊢}; so T_{t}^{⊢}(C_{0}^{⊢}) ⊂ C_{0}^{⊢} is obvious. By the same token t ↦ T_{t}^{⊢}ϕ is continuous for ϕ ∈ A_{0} and then for ϕ ∈ C_{0}^{⊢}. The multiplicativity follows from

(T_{s}^{⊢}(T_{t}^{⊢}ψ))(τ, x) = ∫ (T_{t}^{⊢}ψ)(s + τ, y) T_{s+τ,τ} (x, dy)

= ∫ ∫ ψ(s + t + τ, y′) T_{s+t+τ,s+τ} (y, dy′) T_{s+τ,τ} (x, dy)

= ∫ ψ(t + s + τ, y) T_{s+t+τ,τ} (x, dy) = (T_{s+t}^{⊢}ψ)(τ, x) .