Full Index of Notations

The page where an item is defined appears in boldface. A page number in italics (boldface or no) refers to the Answers in http://www.ma.utexas.edu/users/cup/Answers.

1_{A} = A  the indicator function of A  365
Φ[μ] = μ ∘ Φ^{−1}  the image of the measure μ under Φ  405
A  the idempotent elementary integrands  47
A[F]  the F-analytic sets  432
A_{∞}  the algebra ⋃_{0≤t<∞} F_{t}  22
A_{∞σ}  the countable unions of sets in A_{∞}  35
μ⋆ν  the convolution of μ and ν  413
V^{∗}  dual of the topological vector space V  381
X∗Z  the indefinite Itô integral  133
Z^{⋆}  the maximal process of Z  26
* ϕ, * μ  ϕ , μ reflected through the origin  410
E_{e}^{∗}  the universal completion of E  407
B  the base space [0, ∞) × Ω  22
B̌  product of auxiliary space with base space B  109
η̌ = (η, ϖ)  = (η, s, ω), the typical point of B̌ =^{def} H × B  172
B^{∗}(E) , B^{•}(E)  the Baire & Borel sets or functions of E  391
∥ ∥_{BMO}  the Mean Oscillation Norm  210
B^{•}(E)  the Borel σ-algebra of E  391
B^{•}(ℝ)  the σ-algebra of Borel sets on ℝ  12
b(p)  = \frac{2}{π} ∫_{0}^{∞} ξ^{p−1} sin ξ dξ  458
[[[0, t]]]  =^{def} ℝ_{∗}^{d} × [[0, t]]  181
[[[0, T]]]  =^{def} H × [[0, T]]  172
A^{c}  the complement of A  373
C_{b}(E)  the continuous bounded functions on E  376
C_{0}(B)  the continuous functions vanishing at infinity  366
C_{00}(B)  the continuous functions with compact support  370
μ^{Γ}  the characteristic function of μ for Γ  410
\binom{p}{ν}  p choose ν = p(p − 1) ⋯ (p − ν + 1)/ν!  388
ℂ  the complex plane  363

2 Index of Notations

C_{b}^{k}  the bounded functions with k bounded cont. partials  281
C^{k}(D)  the k-times continuously diff’ble functions on D  372
𝒞^{d}  the path space C_{ℝ^{d}}[0, ∞)  20
𝒞 = 𝒞^{1}  the path space C[0, ∞)  14
𝒞  canonical path space  66
ℭ = ℭ[F.]  the continuous adapted processes  24
δ^{ηθ}  the Kronecker delta  19
δ_{s}  the Dirac measure, or point mass, at s  398
ΔX  the jump process of X ∈ 𝔇 (ΔX_{0} =^{def} 0)  25
𝒟_{E}, 𝒟_{E}^{u}  Skorohod path space, — stopped at u  443
D^{λ}F [u]  the λth (weak) derivative of F at u  305
𝒟_{E}^{u}  the Skorohod paths stopped at u  444
dF  the measure with distribution function F  406
d F = |dF |  its variation  406
𝒟^{(n)}  the path space of the n^{th} random walk  426
𝒟  the right-continuous paths with finite left limits  24
𝔇 = 𝔇[F.]  the càdlàg adapted processes  24
dom(f)  the domain of f  363
𝒟^{d}  canonical path space  66
𝒟, 𝒟^{d}, 𝒟_{E}  canonical space of càdlàg paths  66
𝔼^{x}  𝔼^{x}[F] = ∫ F dℙ^{x}  351
𝔼, 𝔼^{ℙ}  expectation under the prevailing probability, ℙ  32
𝔼[f|Φ], 𝔼[f|Y]  the conditional expectation of f given Φ, Y  407
E = E[F.]  the elementary stochastic integrands  46
E_{1}  the unit ball {X ∈ E : −1 ≤ X ≤ 1} of E  51
E_{+}^{↑}  the pointwise suprema of sequences in E  88
E^{σ}, (E^{d})^{σ}  the sequential closure of E, E^{d}  392
𝔢  the vector lattice of step functions on ℝ_{+}  44
E_{00}  the E-confined functions in E  369
\overline{E}_{00}  the confined uniform closure of E_{00}  370
E_{00}^{σ}  the E-confined functions in E^{σ}  393
∅  the empty set  364
E^{ℙ} = E[F.^{ℙ}]  the elementary integrands of the natural enlargement  57
ϵ  denoting measurability, as in f ∈ F/G  391
ϵ  “is member of” or “is measurable on”  23
≐  denoting equality almost surely or of classes  32
=  denotes near equality and indistinguishability  35
0F  coupling coefficients adjusted so that ^{0}F [0] = 0  272
0C  initial condition adjusted correspondingly  272

Index of Notations 3

F_{+}  the positive elements of F  363
F_{t}  the past or history at time t  21
F_{T}  the past at the stopping time T  28
F.  the filtration {F_{t}}_{0≤t≤∞}  21
F_{.−}  the left-continuous version of F.  123
F_{.+}  the right-continuous version of F.  37
F.^{0}[Z]  the basic filtration of the process Z  23
F.[Z]  the natural filtration of the process Z  39
F.^{0}[𝒟_{E}], F_{.+}^{0}[𝒟_{E}]  basic, canonical filtration of path space  66
F_{.+}^{0}[𝒟_{E}]  the canonical filtration of path space  66
F.[𝒟_{E}]  the natural filtration of path space  67
F_{∞}^{∗}  universal completion of F_{∞}  22
F_{δ}  the intersections of countable subfamilies of F  432
F_{∞}, F_{∞}^{∗}  the σ-algebra ⋁_{0≤t<∞} F_{t}, its universal completion  22
E→S  the operator norm sup{∥T e∥_{S} : ∥e∥_{E} ≤ 1}  388
⌊r⌋  the largest integer n with n ≤ r  79
ĝ = 𝔉[g(x)] , 𝔉^{−1}[.]  Fourier transform and inverse  410
F_{T−}  the strict past of T  120
𝔉[⌈⌈.⌉⌉^{∗}]  the processes finite for ⌈⌈.⌉⌉^{∗}  97
F_{σ}  the unions of countable subfamilies of F  432
F_{σδ}  the intersections of countable subfamilies of F_{σ}  432
F.^{ℙ}, F.^{𝔓}  the ℙ, 𝔓-regularization of F.  37
F.^{𝔓} = F̃.  the (𝔓-) regularization of F.  38
F_{.+}^{𝔓}  the natural enlargement of a filtration  39
G_{δ}  the intersections of countable subfamilies of G  441
G  the open sets of the topological space at hand  441
K_{σ}  the unions of countable subfamilies of K  441
K  the compact sets of the topological space at hand  441
F̃  a predictable envelope of F.  125
F [X]_{.−}  left-continuous version of coupling coefficient  271
G_{A}  the graph of the operator A  464
γ^{(q)}  the symmetric q-stable random variable  458
∥γ^{(q)}∥_{p}  = ( ∫ |γ^{(q)}(x)|^{p} dx)^{1/p}  458
S^{Δ} =^{def} S ˙∪ {Δ}  the one-point compactification of S  374
γ_{t} , γ_{tB}  the centered Gaussian with variance t, tB  419
Ȟ =^{def} H × [0, ∞)  the product of auxiliary space with time  177
h_{0}  prototypical sure Hunt function y ↦ |y|^{2} ∧ 1  180
h′_{0}  y ↦ ∫_{[|ζ|≤1]} |e^{i⟨ζ|y⟩} − 1|^{2} dζ, another one  182

4 Index of Notations

η_{p,q}(I)  a factorization constant  192
I  the identity operator or matrix  463
∫ X dZ  the elementary integral for vectors  56
∫ X dZ  the elementary integral  47
∫_{A} F = ∫ A·F  the integral over the set A of the function F  105
∫ X dZ  the (extended or Itô) integral for vectors  110
∫ X dZ  the (extended or Itô) stochastic integral  99
∫ X dZ  the (extended) stochastic integral revisited  134
∫_{0}^{T} X dZ  its value at T ∈ 𝔗  134
∫ X dZ  the (extended) stochastic integral for vectors  134
X∗Z  the indefinite Itô integral for vectors  134
∫ XδZ  the Stratonovich integral  169
X∘Z  the indefinite Stratonovich integral  169
∫_{0}^{T} G dZ  ∫_{0}^{T} G dZ =^{def} ∫ G dZ^{T}  131
∫_{S+}^{T} G dZ  ∫_{0}^{T} G · ((S, ∞)) dZ  131
ȷ_{Z}  the jump measure of Z  181
H∗ȷ_{Z}  the indefinite integral of H against jump measure  181
K[Z]  ingredient in ∥Z ∥_{K^{q}}  209
L^{∞}  the essentially bounded measurable functions  448
L^{p} = L^{p}(ℙ)  the space of p;ℙ-integrable functions  33
L^{0}, L^{0}(F_{t}, ℙ)  (classes of) measurable a.s. finite functions  33
M_{∞}  the limit of the martingale M at infinity  75
| |_{p} = ∥ ∥_{ℓ^{p}}  |x|_{p} =^{def} ( ∑_{ν} |x^{ν}|^{p})^{1/p}, 0 < p < ∞  364
| |_{∞} = ∥ ∥_{ℓ^{∞}}  |x|_{∞} =^{def} sup_{ν} |x^{ν}|  364
| |  any of the norms | |_{p} on ℝ^{n}  364
⟨x|y⟩  the inner product of vectors x and y  238
| |_{∞p}  |F |_{∞p} =^{def} (∑_{ν}( sup_{η}|F_{η}^{ν}|)^{p})^{1/p}  283
ℓ^{p}  the vectors or sequences x = (x^{ν}) with |x|_{p} < ∞  364
ℓ^{0} =^{def} ℝ^{ℕ}  the Fréchet space of scalar sequences  364
ℒ  the left-continuous paths with finite right limits  24
𝔏 = 𝔏[F.]  the collection of adapted maps Z : Ω → ℒ  24
𝔏^{1}[⌈⌈.⌉⌉^{∗}] & 𝔏^{1}[Z−p]  the ⌈⌈.⌉⌉^{∗}- & Z−p-integrable processes  99
𝔏^{1}[ζ−p] = 𝔏^{1}[\begin{matrix}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{matrix}_{ζ−p}^{∗}] the ζ−p-integrable processes  175
𝔏^{1}[Z−p] = 𝔏^{1}[⌈⌈.⌉⌉_{Z−p;ℙ}^{∗}] the Z−p-integrable processes  99
𝔏^{1}[Z−p]  the Z−p-integrable processes  99
Λ^{⟨q⟩}[Z]  the previsible controller for Z  238
𝔐^{∗}(E) & 𝔐.(E)  the σ-additive & order-continuous measures on E  421
𝔐^{∗}[E]  the σ-additive measures on E  406
⋁  ⋁ F = supremum or span of the family F  22

Index of Notations 5

∧ & ∨  a ∨ b & a ∧ b: smaller & larger of a, b  364
∧  a ∧ b is the minimum of a and b  14
𝔐.(E)  the order-continuous measures on C_{b}(E)  421
∥ ∥_{E}  the quasinorm on the quasinormed space E  381
∥ ∥_{E}  the sup–norm on the space E of functions  188
∥I ∥ = ∥I ∥_{L(E,F)}  = sup{∥I(x)∥_{F} : x ∈ E, ∥x∥_{E} ≤ 1}  381
∥u∥ = ∥u∥_{L(E,F)}  = sup{∥u(x)∥_{F} : x ∈ E, ∥x∥_{E} ≤ 1}  381
mesh(S)  the mesh of a random partition  138
M^{g}  the martingale M.^{g} = 𝔼[g|F.]  72
∥Z ∥_{K^{q}}  = inf {∥g ∥_{L^{q}} : g ∈ K[Z]}  209
∥ ∥_{BMO}  = ∥ ∥_{K^{∞}}  210
∥f ∥_{p}^{∗} = ∥f ∥_{p;ℙ}^{∗}  its mean ∥f ∥_{L^{p}(ℙ)}^{∗} =^{def} (∫^{∗} |f|^{p} dℙ)^{1/p}  452
∥f ∥_{p} = ∥f ∥_{L^{p}(ℙ)}  =^{def} (∫ |f|^{p} dℙ)^{1/p}  33
∥ ∥_{M}  a Picard norm on paths  274
\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−p}  the semivariation for \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{p}  53
∥ ∥_{Z−p}^{∗}  the Daniell extension of ∥.∥_{Z−p}  88
\begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{p}^{∗} = \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{p;ℙ}^{∗}  its mean \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{L^{p}(ℙ)}^{∗} =^{def} (∫^{∗}|f|^{p} dℙ)^{1/p ∧1}  452
\begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{p} = \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{L^{p}(ℙ)}  = \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{L^{p}(ℙ)} =^{def} (∫ |f|^{p} dℙ)^{1/p ∧1}  33
\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}  a subadditive mean  95
ζ^{h,t} _{I^{p}}= ζ^{h,t} _{I^{p}[ℙ]}  integrator (quasi)norms of the random measure ζ  173
Z _{I^{p}}= Z _{I^{p}[ℙ]}  integrator (quasi)norms of the integrator Z  55
\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−p}^{∗} = \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−p;ℙ}^{∗}  the Daniell mean  88
\underline{m}[f] = \underline{m}[f; ξ′]  local constant for a single-step method ξ′  281
Z _{I^{p}} = Z _{I^{p}[ℙ]}  =^{def} sup {∥∫ X dZ ∥_{L^{p}(ℙ)} : X ∈ E , |X| ≤ 1}  55
Z _{I^{p}} = Z _{I^{p}[ℙ]}  =^{def} sup {∥∫ X dZ ∥_{L^{p}(ℙ)} : X ∈ E , |X| ≤ 1}  56
∥ ∥_{[α]}^{∗}  the corresponding mean  452
∥ ∥_{[α]}  ∥f ∥_{[α]} =^{def} inf{λ > 0 : ℙ[|f| > λ] ≤ α}  34
∥ ∥_{Z−[α]}  the corresponding semivariation  53
∥ ∥_{Z−[α]}^{∗}  the Daniell extension of ∥.∥_{Z−[α]}  88
[α]  the integrator size according to ∥ ∥_{[α]}  55
p,M, _{p,M}^{⋆}  Picard Norms  283
μ_{Z}  the Doléans–Dade measure of Z  222
\begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{0} = \begin{aligned}⌈ \\ ⌈ & f & ⌉ \\ & & ⌉ \end{aligned}_{0;ℙ}  the metric inf\{λ : ℙ[|f| ≥ λ] ≤ λ\} on L^{0}  34
F^{μ}  μ-completion of F  414
N(0, t)  the law “normal zero–t”  419
O( . ), o( . )  big O and little o  388
(Ω, F.)  the underlying filtered measurable space  21

6 Index of Notations

O  σ-algebra of optional or well-measurable sets  440
ϖ  the typical point (s, ω) of ℝ_{+} × Ω  22
ℙ  the pertinent probability on (Ω, F)  3
𝔓  the pertinent probabilities  32
P_{00}  the bdd. predictable processes with bdd. carrier  128
P_{b}  the bounded predictable processes  135
𝔓[Z]  the probabilities for which Z is an L^{0}-integrator  61
P̌ =^{def} B^{•}(H) ⊗ P  = (E[H]⊗E[F.])^{σ}, predictable random functions  172
𝔓^{∗}(E) = 𝔐_{1,+}^{∗}(E)  the probabilities on C_{b}(E)  421
𝔓.(E) = 𝔐._{1,+}(E)  the order-continuous probabilities on E  421
p^{⋄} = p^{⋄}[Z]  p if Z jumps, 2 otherwise  238
1^{⋄} = 1^{⋄}[Z]  2 if Z is a martingale, 1 otherwise  238
P = P[F.]  the predictable processes or σ-algebra  115
P^{ℙ}  the processes previsible with ℙ  118
ℙ_{t}  the restriction of ℙ to F_{t}  40
ℚ  the rationals  363
ℚ^{t}  =^{def} {q ∈ ℚ : 0 ≤ q < t} ∪ {t}  26
ℝ_{+}  the positive reals, i.e., the reals ≥ 0  363
ℝ_{∗}^{d}  punctured d-space ℝ^{d} \ {0}  363
T_{A}  the reduction of T ∈ 𝔗 to A ∈ F_{T}  31
0_{A}  the stopping time 0 reduced to A ∈ F_{0}  48
\overline{ρ}(r, s)  the arctan metric on \overline{ℝ}  364
R^{l}F [u; v]  Taylor Remainder of degree l of F as v → u  305
\overline{ℝ}  the extended reals \{−∞\} ∪ ℝ ∪ \{∞\}  363
ℝ  the reals  363
σ, E^{σ}, Eσ ; \overline{ℝ}  ^{σ} denotes sequential closure, of E  392
sgn  sgn z = 1, 0, −1 according as z > 0, = 0, < 0  388
sign  signx = 1, −1 according as x ≥ 0, < 0  330
𝔰_{M}  Banach space of paths with ∥ ∥_{M} < ∞  275
cV,^{j}V  continuous & jump part of finite variation process V  69
c̃Z,^{r}Z  continuous martingale part of Z, rest Z −^{c̃}Z  235
s̃Z &^{l}Z  small-jump martingale part & large-jump part of Z  237
v̂Z  a continuous finite variation part of Z  237
pZ  the part of Z supported on a sparse previsible set  235
qZ  the quasi-left-continuous rest Z −^{p}Z  235
[Z, Z],^{c}[Z, Z] &^{j}[Z, Z]  square bracket, its continuous & jump parts  148
[Y, Z],^{c}[Y, Z] &^{j}[Y, Z]  square bracket, its continuous & jump parts  150
c[Y, Z]  continuous part of the square bracket  150
j[Y, Z]  jump part of the square bracket  150

Index of Notations 7

𝔖_{p,M}^{n}  processes with finite Picard Norm _{p,M}  283
𝔖_{p,M}^{⋆n}  processes with finite Picard Norm _{p,M}^{⋆}  283
S[Z] & σ[Z]  square function & continuous square function of Z  148
σ[Z]  continuous square function of Z  148
S  Schwartz space of C^{∞}-functions of fast decay  269
σ(C_{b}^{∗}(E), C_{b}(E))  the topology of weak convergence of measures  421
Z^{t}  Z_{s}^{t} = Z_{s∧t} is the process Z stopped at t  23
Z^{T}  Z_{s}^{T} = Z_{T ∧s} is the process Z stopped at T ∈ 𝔗  28
σ(V, M)  topology generated on V by the functions M  381
Z^{S}  the S-scalæfication of Z  139
[Z, Z]  the square bracket of Z  148
c[Z, Z],^{j}[Z, Z]  its continuous & jump parts  148
j[Z, Z]  the jump part of the square bracket of Z  148
T^{l}F [u]  Taylor polynomial of degree l of F at u  305
𝔗 = 𝔗[F.]  the F.-stopping times  27
[[T]] = [[T, T]]  the graph of the stopping time T  28
T. : λ ↦ T^{λ}  the time transformation for Z  239
T_{p,q}(.)  type of a map or space  461
U = U_{α}(C_{0}(E))  the range of the resolvent operators  463
X_{.−} & X_{.+}  left- & right-continuous version of X  24
X_{.+}  the right-continuous version of X  24
V^{μ}  the Doléans–Dade process of μ  222
Λ^{⟨q⟩}[Z]  a previsible fin. var. process controlling Z  245
Λ^{⟨q⟩}[ζ]  a previsible controller  251
z , μ  variation of distribution function z, or measure μ  45
dz = |dz|  the variation measure of the measure dz  45
ĵ_{Z} & j̃_{Z}  compensator & compensatrix of jump measure  232
j̃_{Z}  compensated part of the jump measure  232
𝕎  Wiener measure  16
\underline{W}  Wiener process as a C[0, ∞)-valued random variable  14
F.^{0}[W.]  the basic filtration of Wiener process  18
ξ.^{f}(x) = ξ(x,. ; f)  the flow generated by f  278
Z^{[ν]}  the higher order brackets  239
Z^{⟨ρ⟩}  higher order previsible brackets  240
ḟ  the equivalence class of f mod negligible functions  13
Z_{∞} = Z_{∞−}  the limit (possibly ±∞) of Z at ∞  27
≪. & ≈.  local absolute continuity & local equivalence  40
≪  denoting absolute continuity  407
≈.  denoting local equivalence  40

8 Index of Notations

\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{L^{p}(ℙ)}^{∗}  the corresponding mean  452
Z.(ω)  the path s ↦ Z_{s}(ω)  23
⇒  denoting weak convergence of measures  421
Z_{T}  the random variable ω ↦ Z_{T (ω)}(ω).  28
Z_{s}  the function ω ↦ Z(s, ω)  23
ζ^{T} = [[[0, T]]]ζ  the random measure ζ stopped at T  173
a ∼ b  means both a/b and b/a are bounded by a constant < ∞ 218
ζ̂, ζ̃  previsible & martingale parts of random measure ζ  231
ζ̃  martingale part of the random measure ζ  231
Ẑ, Z̃  previsible & martingale parts of the integrator Z  221
ζ[X]  lifetime of the solution X  273

Symbols

Full Index

The page where an item is defined appears in boldface. A page number in italics (boldface or no) refers to the Answers in http://www.ma.utexas.edu/users/cup/Answers.

A absolute-homogeneous, 33, 54, 380 absolutely continuous, 407

locally, 40 absorb, 379, 451, 465, 513 accessible stopping time, 122 action, differentiable, 279, 326 adapted, 23, 25

map of filtered spaces, 64, 317,

348 process, 23, 28

adaptive, 7, 141, 281, 317, 319 adjusted, initial condition ^{0}C, 272

coupling coefficient ^{0}F , 272, 286 a.e., almost everywhere, 32, 96

meaning Z-a.e., 129 a.e.,

meaning Z-almost everywhere,

129 Alaoglu, 425 algebra, 366 algorithm,

computing integral pathwise, 140 solving a markovian SDE, 311 solving a SDE pathwise, 7, 314

a.s., almost surely, ℙ-a.s., 32 almost everywhere, 96 ambient set, 90 analytic set, 432 analytic space, 441 angle bracket ⟨ , ⟩, 228 approximate identity, 447, 527 approximation, numerical – see

method, 280 arbitrarily large,

stopping times, 51 arc, 153 arctan metric, 110, 364, 369, 375 Ascoli–Arzela theorem, 7, 385, 428 ASSUMPTIONS,

on the data of a SDE, 272 the filtration is natural (regular and right-continuous), 58, 118, 119, 123, 130, 162, 221, 254, 437 the filtration is right-continuous,

140 atomic decay, 122 atom of an algebra of sets, 391 autologous,

coupling coefficient, 288, 299, 305 auxiliary base space, 172 auxiliary space, 171, 172, 239, 251 average, 504

B

background noise, 2, 8, 272

stationary, 10 backward equation, 359 Baire & Borel, function, set & σ-algebra, 391, 393 Baire & Borel functions, 391, 393 ball, 377, 379 Banach space,

10 Index

-valued integrable functions, 409 base space,

auxiliary, 172 , 22, 40, 64, 90, 172

basic filtration, 18 of a process, 19, 23 of Wiener process, 18 on path space, 66, 445

basis, stochastic, 22 of neighborhoods, 374 of a uniformity, 374

Bernoulli random variables, 426, 455 bigger than (≥), 363 Big O, 388 Blumenthal, 352 Bochner integrable, 400, 463 Bochner’s theorem, 458 Borel,

σ-algebra, 391 Borel function, 391 Borel σ-algebra on path space, 15 bounded,

I^{p}[ℙ]-bounded process, 49 linear map, 452 polynomially, 322 process of — variation, 67 stochastic interval, 28 subset of a topological vector

space, 379, 451 subset of L^{p}, 451, 513 variation, 45

bounded away from zero, 182 boundedawayfromzero, 208 bounded away from zero, 356, 367,

416 bounded graph,

stopping time with —, 69 bounded mean oscillation, 210 (B-p), 50, 53 bracket,

previsible, oblique or angle —,

228 square —, 148, 150 brackets,

higher order, 239

higher order previsible, 240 Brown, 9 Brownian motion, 9, 12, 18, 80 Brownian sheet, 20 Burkholder, 84, 213

C càdlàg, càglàd, 24 calculus, 141, 168, 241 canonical,

decompositions of an integrator,

221, 234, 235 filtration on path space, 66, 335 Wiener process, 16, 58

canonical representation, 66 of an integrator, 14, 67 of a random measure, 177 of a semigroup, 358

capacitable, 434 capacity, 124, 252, 398, 434 Caratheodory, 395 carrier of a function, 369 Cauchy,

distribution, 458

filter, 374

random variable & distribution,

458 Cauchy–Schwarz inequality, 150 Central Limit Theorem, 10, 423, 578 Chapman–Kolmogorov equations,

465 characteristic function, 13, 161, 254,

410, 430, 458, 507 of a Gausssian, 419 of a Lévy process, 263 of Wiener process, 161

characteristic triple, 232, 259, 270 chopping, 47, 366

closed under —, 47, 90, 94, 95,

105, 366, 395 Choquet, 435, 442 class C_{b}^{k} etc, 281 closed,

operator, 464

sequentially, 391

set, 373 closure,

sequential, 392, 537

topological of a set, 373 cofinal subset, 416 collor, corlol, 24 commuting flows, 279 commuting vector fields, 279, 326 commuting with translation, 269 compact, 373, 375

paving, 432 compatible,

integrator — with a time trans-

formation, 553 compensated part of an integrator,

221 compensator, 185, 221, 230, 231 compensatrix, 221 complete,

uniform space, 374

universally — filtration, 22 completely regular, 373, 376, 421 completion,

of a filtration, 39

of a uniform space, 374

universal, 22, 26, 407, 436

with respect to a measure, 414 E-completion, 369, 375 completely (pseudo)metrizable, 375 compound Poisson process, 270 computer, 140, 311, 536 conditional expectation, 5, 71, 72,

349, 408, 511 condition(s),

(IC-p), 89

(B-p), 50, 53

(RC-0), 50

the natural, 39 confined, 369, 394 confined (cont’d) E-confined, 369 self-confined, 172, 393 uniform limit, 370

conjugate exponents, 76, 449 conservative,

generator, 466, 467

semigroup, 268, 465, 466, 467 consistent family, 164, 166, 401

full, 402

of probabilities, 401 constant,

ultimately, 119 continuity, 373 continuity along increasing se-

quences, 95, 124, 137, 398, 452 of a capacity, 434 of E, 89 of E_{+} , 94 of E, 106 of E_{+}^{↑}, 90, 95, 125

Continuity Theorem, 254, 423 continuous, process, 23

continuous martingale part, of a Lévy process, 258 of an integrator, 235 continuous semigroup, 463 contraction semigroup, 463 contractivity, 235, 251, 407, 415 modulus, 274, 290, 299, 407 strict, 274, 275, 282, 290, 291, 297,

407 control,

previsible, of integrators, 238 previsible, of random measures,

251 sure, of integrators, 292

controller, the, for random measures, 251 the previsible, 238, 283, 294

control measure, 129 CONVENTIONS, 21 CONVENTIONS (cont’d)

about ±∞, 363 about positivity, negativity etc.,

363 = denotes near equality and in-

distinguishability, 35 Einstein convention, 8, 146 Z-almost everywhere, 129 Z-measurable, 129 Z-negligible, 129 ∈ meaning “measurable on”, 23,

391 ∫_{0}^{T} G dZ = ∫ G dZ^{T} , 131 ∫_{0}^{T} G dZ is the random variable

(G∗Z)_{T} , 134 integrators have right-continuous

paths with left limits, 62 ℙ is understood, 55 ΔX_{0} = X_{0} , 25 X_{0−} = 0, 24 processes are only a.e. defined, 97 re Z-negligibility, Z-measurability,

Z-a.e., 129 sets are functions, 364 [Y, Z]_{0} = ^{j}[Y, Z]_{0} = Y_{0}·Z_{0} , 150 [Z, Z]_{0} = ^{j}[Z, Z]_{0} = Z_{0}^{2} , 148

convergence,

\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-a.e. & Z−p-a.e., 96

in law or distribution, 421 in mean, 6, 96, 99, 102, 103, 449 in measure or probability, 34 in probability, 34 largely uniform, 111 of a filter, 373 uniform — on large sets, 111 weak, of measures, 421

convex, function, 147, 408, 514 function of an L^{0}-integrator, 146 set, 146, 188, 197, 379, 382 strictly — function, 408

convolution, 117, 196, 199, 275, 288, 373, 413 convolution (cont’d) semigroup, 254

core, 270, 464

countably generated, algebra or vector lattice, 367 , 100, 377, 566, 568

countably subadditive, 87, 90, 94,

396 counting measure, 180, 413, 556 coupling coefficient,

autologous, 288, 299, 305

autonomous, 287 , 1, 2, 4, 8, 272 depending on a parameter, 302,

305 endogenous, 289, 314, 316, 331 instantaneous, 287 Lipschitz, 285 Lipschitz in p-mean, 286 Lipschitz in Picard norm, 286 markovian, 272, 321 non-anticipating, 272, 333 of linear growth, 332 randomly autologous, 288, 300 strongly Lipschitz, 285, 288, 289

Courrège, 232 covariance,

Wiener process with —, 161, 258 covariance matrix, 10, 19, 161, 420 cross section, 204, 331, 378, 436, 438,

440 cumulative, noise, 2

cumulative distr. function, 69, 406 cylinder function, 354, 401

D

Daniell mean, the, 89, 95, 109, 124, 174 , 87, 397 maximality of, 123, 135, 217, 398

Daniell’s method, 87, 395

Davis, 213 DCT: Dominated Convergence The-

orem, 5, 52, 103 debut, 40, 127, 437 decompositions, canonical, ,

of an integrator, 221, 234, 235 of a Lévy process, 265

decreasingly directed, 366, 398 Dellacherie, 432 dense,

set in a topological space, 373 topology, 415

density, 407, 410 derivative, 388

higher order weak, 305

tiered weak, 306

weak, 302, 390 deterministic, not depending on

chance ω, 7, 47, 123, 132 differentiability,

Fréchet–, 555 differentiable,

action, 279, 326

, 388

l-times weakly, 305

weakly, 278, 390 differential form,

of Itô’s Formula, 158 diffusion coefficient, 11, 19 Dirac measure, 41, 398 discrete, 31 disintegration, 231, 258, 418 dissipative, 464 distinguish processes, 35 distribution,

cumulative — function, 406 , 10, 14, 405 Gaussian or normal, 419 normal, 419

distribution function, 43, 45, 87, 263,

406 of Lebesgue measure, 51 random, 49 stopped, 131 Doléans–Dade,

exponential, 159, 163, 167, 180,

186, 219, 326 measure, 222, 226, 241, 245, 247,

248, 249, 251, 547 process, 222, 226, 231, 245, 247,

249 Donsker, 426 Doob, 60, 75, 76, 77, 223, 521, 529 maximal lemma of —, 76 optional stopping theorem of —,

77, 521 Doob–Meyer decomposition, 187,

202, 221, 228, 231, 233, 235, 240, 244, 247, 254, 266

δ-ring, 104, 394, 409 drift, 4, 72, 272 drive, 9 driver, 11, 20, 169, 188, 202, 228, 246 of a stochastic differential equa-

tion, 4, 6, 17, 271, 311 driving term, 4, 11 d-tuple of integrators, see vector of —, 9

dual of a topological vector space,

381 dyadic rationals, 36, 38, 385

E

effect, of noise, 2, 8, 272 Einstein convention, 146 elementary,

integral, 7, 44, 47, 99, 395

integrand, 7, 43, 44, 46, 47, 49, 56, 79, 87, 90, 95, 99, 111, 115, 136, 155, 395

stochastic integral, 47 stochastic integrand, 46 stopping time, 47, 61

endogenous coupling coefficient, 289, 314, 316, 331 enlargement,

natural, of a filtration, 39, 57, 101,

440 usual, of a filtration, 39

envelope, 125, 129, 135 Z-envelope, 129 measurable, 398 predictable, 125, 129, 135, 337 predictable \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-—, 125

equicontinuity, uniform, 384, 387 equivalence class ḟ of a function, 32 equivalent, locally, 40, 162 equivalent probabilities, 32, 34, 50,

187, 191, 206, 208, 450 ESP, extrasensory perception, 35 essence, 172, 175, 176 Euler–Peano approximation, 7, 311,

312, 314, 317 evaluation map, 15 evaluation process, 66 evanescent, 35 event, 3 evolution equation, 342 evolution of the world, 3 exceed, 363 expectation, 32, 505 exponential, stochastic, 159, 163,

167, 180, 186, 219 extended reals, 23, 60, 74, 90, 125,

363, 364, 375, 392, 394, 408 extension, integral, of Daniell, 87

F factorization, 50, 192, 233, 282, 291,

293, 295, 298, 304, 320, 458, 523, 564

of random measures, 187, 208, 297 factorization constant, 192 Fatou’s lemma, 449, 539 Fefferman, 213 Fefferman’s inequality, 212 Feller semigroup,

canonical representation of, 358 Feller semigroup (cont’d)

conservative, 268, 465 convolution, 268, 466 , 269, 350, 351, 465, 466 natural extension of, 467 stochastic representation, 351 stochastic representation of, 352

Feynman-Kac formula, 359 filter, convergence of, 373 filter, 373 filtered measurable space, 21, 438 filtration,

basic, of a process, 18, 19, 23, 254 basic, of path space, 66, 166, 445 basic, of Wiener process, 18 canonical, of path space, 66, 335 , 5, 22 full, 166, 176, 177, 185 full measured, 166 𝔓-regular, 38, 437 is assumed right-continuous and

regular, 58, 60, 62 measured, 32, 39 natural enlargement of, 39, 57,

254 natural, of a process, 39 natural, of a random measure, 177 natural, of path space, 67, 166 regular, 38, 437 regularization of a, 38, 135 right-continuous, 37 right-continuous version of, 37, 38,

117, 438 universally complete, 22, 437, 440

finite, for the mean, 90, 94 function — in p-mean, 448 process — for the mean, 97, 100 process of — variation, 68 stochastic interval, 28 variation, 45 variation, process of —, 23

finite intersection property, 432 finite variation, 394, 395 flow, 278, 326, 359, 466 λ-form, 305 Fourier inversion formula, 411 Fourier transform, 410 Fréchet space, 364, 380, 400, 467 Fréchet differentiability, 555 French School, 21, 39 Friedman, 459 F.-stopping time, 27 Fubini’s theorem, 403, 405 full,

filtration, 166, 176, 185 measured filtration, 166 projective system, 402, 447

function, Baire & Borel, 391, 393 Banach space-valued integrable,

409 finite in p-mean, 448 p-integrable, 33 idempotent, 365 lower semicontinuous, 194, 207,

376, 393 numerical, 110, 364 simple measurable, 448 universally measurable, 22, 23,

351, 407, 437 upper semicont., 108, 194, 207,

376, 382 vanishing at infinity, 366, 367

functional, 370, 372, 379, 381, 394 functional, solid, 34 Fundamental Theorem of Calculus,

157, 170, 304, 401, 464

G Gamma function, 420 Garsia, 215 gauge, 50, 89, 95, 130, 380, 381, 400,

450, 468 gauges defining a topology, 95, 379 Gauß kernel, 420 Gaussian,

distribution, 419

normalized, 12, 419, 423

semigroup, 466

with covariance matrix, 270, 420 G_{δ}-set, 379 Geiger counter, 122 Gelfand transform, 108, 174, 370,

535, 567 generate,

a σ-algebra from functions, 391 a topology, 376, 411

generator, conservative, 467 generator of a semigroup, 351, 463 Girsanov’s theorem, 39, 162, 164,

167, 176, 186, 338, 475 global L^{p}(ℙ)-integrator, 50 global L^{p}-random measure, 173 graph of an operator, 464 graph of a random time, 28 grave, 374, 465 Gronwall, 383 Gundy, 213

H Haagerup, 457 Haar measure, 196, 394, 413 Hahn–Banach theorem, xii, 194, 301,

523, 567 Hardy mean, 179, 217, 219, 232, 234,

250 Hardy space, 220 harmonic measure, 340 Hausdorff space or topology, 367,

370, 373 heat kernel, 420 Heun method, 322 heuristics, 10, 11, 12 history,

course of, 3, 318

, 5, 21 hitting distribution, 340 Hölder’s inequality, 33, 188, 449 homeomorphism, 364, 379, 441, 460 Hunt function, 181, 185, 257, 258 C^{∗}-algebra, 504

I (IC-p), 89 idempotent function, 365 identity operator, 463 iff, meaning if and only if, 37 L^{p}(ℙ)-integrator, 50

global, 50

vector of, 56 i.i.d.=independent identically distri-

buted, 79 image of a measure, 405 improper Riemann integral, 463 increasingly directed, 72, 366, 376,

398, 417 increasing process, 23 increments, 10, 19

independent, 10, 253 indefinite integral,

against an integrator, 134 against jump measure, 181

independent, increments, 10, 253 , 10, 16, 263, 413

indistinguishable, 35, 62, 226, 387 induced topology, 373 induced uniformity, 374 inequality,

Cauchy–Schwarz —, 150 Hölder’s —, 449 of Hölder, 33, 188

initial condition, 8, 273 initial value problem, 279, 280, 342,

464, 556 inner measure, 396 inner regularity, 400 instant, 31 instantaneous,

coupling coefficient, 287 integrability criterion, 113, 397 integrable,

Bochner —, 400, 463

function, Banach space-valued,

409 process, on a stochastic interval,

131 set, 104 integral,

against a random measure, 174 elementary, 7, 44, 47, 99, 395 elementary stochastic, 47 extended, 100 extension of Daniell, 87 Itô stochastic, 99, 169 stochastic, 99 Stratonovich stochastic, 169, 320 upper, of Daniell, 87 Wiener, 5 integral equation, 1 integrand,

elementary, 7, 43, 44, 46, 47, 49, 56, 79, 87, 90, 95, 99, 111, 115, 136, 155

elementary stochastic, 46

, 43 integrator,

d-tuple of, see vector of, 9

, 7, 20, 43, 50 local, 51, 234 slew of, see vector of, 9

integrators,

vector of, 56, 63, 77, 134, 149, 187,

202, 238, 246, 282, 285 intensity, 231 measure, 231 of jumps, 232, 235 rate, 179, 185, 231 interior of a set, 373 interpolation, 33, 85, 453, 517 interval,

finite stochastic, 28 random —, 28 stochastic, 28 intrinsic time, 239, 318, 321, 323 invariant, 464 involution, 504 iterates,

Picard, 2, 3, 6, 273, 275 Itô, 5, 24 Itô’s theorem, 157 Itô stochastic integral, 99, 169

J Jensen’s inequality, 73, 243, 408, 515 jump intensity,

continuous, 232, 235

, 232

measure, 258

rate, 232, 258 jump measure, 181 jump process, 148

K Khintchine, 64, 455 Khintchine constants, 457 Kolmogorov, 3, 384 Kronecker delta, 19 Kunita–Watanabe, 212, 228, 232 Ky–Fan, 34, 189, 195, 207, 382

L lacunary, 455 Laplace transform,

of a semigroup, 463 largely,

smooth, 111, 175

uniform convergence, 111

uniformly continuous, 405 lattice algebra, 366, 370 law, 10, 13, 14, 161, 405

Newton’s second, 1

of Wiener process, 16

strong law of large numbers, 76,

216 symmetric stable, 458

Lebesgue, 87, 395 Lebesgue measure, 394 Lebesgue–Stieltjes integral, 4, 43, 54, 87, 142, 144, 153, 158, 170, 232, 234 left-continuous, process, 23 time transformation, 550 version of a process, 24 version of the filtration, 123 ℓ^{p}-length of a vector or sequence,

364 less than (≤), 363 Lévy,

measure, 258, 269

process, 239, 253, 255, 292, 349,

466 Lévy’s characterization of Wiener

process, 19, 160 Lie bracket, 278 lifetime of a solution, 273, 292 lifting, 415 strong, 419

Lindeberg condition, 10, 423 linear functional, positive, 395 linear map, bounded, 452 Lipschitz,

constant, 144, 274

coupling coefficients, 285

function, 2, 4, 144 in p-mean, 286 in Picard norm, 286 locally, 292 strongly, 285, 288, 289 vector field, 274, 287

Little o, 388

local, L^{1}-integrator, 221 L^{p}-random measure, 173 martingale, 75, 84, 221 property of a process, 51, 80 local E-compactification, 108, 367,

370 local martingale, 75 local martingale (cont’d)

square integrable, 84 local time, 147 locally absolutely continuous, 40, 52,

130, 162 locally bounded, 122, 225 locally compact, 374 locally convex, 50, 187, 379, 467 locally equivalent probabilities, 40,

162, 164 locally Z−p-integrable, 131 locally Lipschitz, 292 logcharacteristic function, 256, 257 Lorentz space, 51, 83, 452 lower integral, 396 lower semicontinuous,

function, 194

, 207, 376, 393 L^{p}-bounded, 33 L^{p}(ℙ)-integrator, 50 L^{p}-integrator, 50, 62

complex, 152

vector of, 56 Lusin, 110, 432

space, 447

space or subset, 441 L^{0}-integrator,

convex function of a, 146

M (M), condition, 91, 95 markovian coupling coefficient, 272,

287, 311, 321 Markov property, 349, 352

strong, 349, 352 martingale,

continuous, 161

locally square integrable, 213 local, 75, 84 , 5, 19, 71, 74, 75 of finite variation, 152, 157 previsible, 157 previsible, of finite variation, 157 martingale (cont’d)

square integrable, 78, 163, 186,

262 uniformly integrable, 72, 75, 77,

123, 164 martingale measure, 220 martingale part, continuous, of an

integrator, 235 martingale problem, 338 martingale random measure, 180 ℙ-martingale, 77 Maurey, 195, 205 maximal inequality, 61, 337 maximality,

of a mean, 123

of Daniell’s mean, 123, 135, 217,

398 maximal lemma of Doob, 76 maximal path, 26, 274 maximal process, 21, 26, 29, 38, 61, 63, 122, 137, 159, 227, 360, 443, 544

maximum principle, 340

MCT: Monotone Convergence Theo-

rem, 102 mean,

the Daniell —, 89, 124, 174 controlling a linear map, 95 controlling the integral, 95, 99,

100, 101, 123, 124, 217, 247, 249, 250

Hardy—, 217, 219, 232, 234, 250 σ-finite –, 105 majorizing a linear map, 95 maximality of a —, 123 maximality of Daniell’s —, 123,

135, 217 , 94, 399 pathwise, 216 measurability, 111 measurable,

cross section, 436, 438, 440 filtered — space, 21, 438 measurable (cont’d)

Z-measurable, 129 Z−p-measurable, 111 , 110 on a σ-algebra, 391 process, 23, 243 set, 114 simple function, 448 space, 391

measure, convergence in —, 34 σ-finite, 449 , 394 of totally finite variation, 394 support of a, 400 tight, 21, 165, 399, 407, 425, 441

measured filtration, 39 measure space, 409 mesh of a random partition, 138 method,

adaptive, 281 Euler, 280, 281, 311, 314, 318, 322 Heun, 322 order of, 281, 324 Runge–Kutta, 282, 321, 322 scale-invariant, 282, 329 single-step — ξ′, 280 single-step — Ξ′, 322, 325 Taylor, 281, 282, 321, 322, 327

metric, 374 metrizable, 367, 375, 376, 377 Meyer, 228, 232, 432 minimax theorem, 382 Minkowski functional, 380 μ-integrable, 397 μ-measurable, 397 μ-negligible, 397 model, 1, 2, 4, 6, 8, 9, 10, 11, 18, 20,

21, 27, 32 modification, 58, 61, 62, 75, 101, 132,

134, 136, 147, 167, 223, 268 ℙ-modification, 34, 384 modulus,

of continuity, 58, 191, 206, 209,

444, 535 of contractivity, 290, 299, 407

momentum, 1

monotone class, 16, 145, 155, 218,

224, 393, 439, 521, 531, 532, 574 Monotone Class Theorem, 16 morphism, 66 morphism of filtered spaces, 64 multiplicative class,

complex, 393, 399, 410, 421, 423,

425, 427 , 393, 399, 421 μ-measurable,

map between measure spaces, 405 mutare, to change, 129, 165, 251, 406

N

natural, domain of a semigroup, 467 extension of a semigroup, 467 filtration of path space, 67, 166 increasing process, 228, 234 natural conditions, 29, 39, 60, 118,

119, 127, 130, 133, 162, 227, 437 natural enlargement of a filtration,

57, 101, 254, 440 nearly, ℙ-nearly, 35, 38, 358, 511 nearly, 38, 52, 58, 71, 74, 75, 118,

119, 134, 141, 146, 152, 158, 304, 551

nearly empty set, 35, 36, 60, 62, 127, 136, 150, 154, 167, 183, 283, 295, 355, 518, 520, 533, 553

ℙ-nearly, 35, 38 negative, (≤ 0), 363

strictly — (< 0), 363 negligible,

meaning Z-negligible, 129

, 32, 95 ℙ-negligible, 32 neighborhood filter, 373 Nelson, 10 Newton, 1 Nikisin, 207 noise,

background, 2, 8, 272

cumulative, 2

effect of, 2

, 2 nolens volens, meaning nilly–willy,

421 non-adaptive, 281, 317, 321 non-anticipating,

coupling coefficient, 333 process, 6, 144 random measure, 173

non-increasing rearrangement, 451 normal distribution, 419 normal distribution with covariance

matrix, 420 normalized Gaussian, 12, 419, 423 normed algebra, 504 Notations, see Conventions, 21 Novikov, 163, 165 numerical,

function, 110, 364

, 23, 392, 394, 408 numerical approximation, see

method, 280

O observable, 504 ODE, 4, 273, 326 one-point compactification, 356, 366,

374, 465 opaque, 162 open set, 373 operator norm, 388, 463 optional,

σ-algebra O, 440

process, 440

projection, 440

stopping theorem, 77, 521 order-bounded, 406 order-complete, 252, 406 order-continuity, 398, 421 order-continuous mean, 399 order interval, 45, 50, 173, 372, 449,

513 order of an approximation scheme,

281, 319, 324 oscillation of a path, 444 outer measure, 32, 396

P

partial derivative, 389

partition, random or stochastic, 138 partition,

random, 138 stochastic, 138

past, of a stopping time, 28 , 5, 21 strict, of a stopping time, 120,

225, 530 path, 4, 5, 11, 14, 19, 23

—s describing the same arc, 153 path space,

basic filtration of, 66

canonical filtration of, 66, 335 canonical, 66 natural filtration of, 67, 166 , 14, 20, 166, 380, 391, 411, 440

pathwise, computation of the integral, 140 failure of the integral, 5, 17 mean, 216, 238 previsible control, 238 solution of an SDE, 311 stochastic integral, 7, 140, 142,

145 pauculi: a very few, 110, 174 paucity, 110, 174 paved set, 432 paving,

compact, 432

, 432 permanence properties,

of fullness, 166 of integrable functions, 101 of integrable sets, 104 of measurable functions, 111, 114,

115, 137 of measurable sets, 114 , 165, 391, 397

phase space, 1, 10 Picard,

iterates, 2, 3, 6, 273, 275 iteration, 1, 313 , 4, 6

Picard norm, Lipschitz in, 286 , xii, 274, 283

p-integrable, function or process, 33 process, 33 uniformly — family, 449

Pisier, 195 p-mean, 33 p-mean σ-additivity, 106 p;ℙ-mean σ-additivity, 90 point at infinity, 374 point process, 183

Poisson, 185 simple, 183

Poisson point process, compensated, 185 , 185, 264

Poisson process, 268, 270, 359 Poisson random variable, 420 Poisson semigroup, 359 polarization, 366 polish space, 15, 20, 440 polynomially bounded, 322 positive (≥ 0),

linear functional, 395 , 363 strictly — (> 0), 363

positive maximum principle, 269, positive semidefinite,

inner product, 150 matrix, 161, 258, 259, 420, 539

𝔓-regular filtration, 38, 437 precompact, 376 predictable,

envelope, 125, 129, 135, 337 increasing process, 225 , 115 process of finite variation, 117 process, 115, 125 projection, 439 random function, 172, 175, 180 stopping time, 118, 438 transformation, 185

predict a stopping time, 118 prelocal, xii previsible,

bracket, 228 control, 238 dual — projection, 221 , 68 process of finite variation, 221 process, 68, 122, 138, 149, 156,

217, 228, 238 process — with ℙ, 118 set, 118 set, sparse, 235 square function, 228

previsible controller, 238, 283, 294 probabilities,

locally equivalent, 40, 162 probability,

convergence in —, 34 on a topological space, 421 , 3, 505

process, adapted, 23 basic filtration of a, 19, 23, 254 continuous, 23 defined \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-a.e., 97 evanescent, 35 finite for the mean, 97, 100 process (cont’d)

I^{p}[ℙ]-bounded, 49 L^{p}-bounded, 33 p-integrable, 33 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-integrable, 99 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-negligible, 96 Z−p-integrable, 99 Z−p-measurable, 111 increasing, 23 indistinguishable —es, 35 integrable on a stochastic interval,

131 jump part of a, 148 jumps of a, 25 left-continuous, 23 Lévy, 239, 253, 255, 292, 349 locally Z−p-integrable, 131 local property of a, 51, 80 maximal, 21, 26, 29, 61, 63, 122,

137, 159, 227, 360, 443, 544 measurable, 23, 243 modification of a, 34 natural increasing, 228 non-anticipating, 6, 144 of bounded variation, 67 of finite variation, 23, 67 optional, 440 predictable increasing, 225 predictable of finite variation, 117 predictable, 115, 125 previsible of finite variation, 221 previsible, 68, 118, 122, 138, 149,

156, 217, 228, 238 previsible with ℙ, 118 , 6, 23, 90, 97 reduced to a property at a stop-

ping time, 51 right-continuous, 23 square integrable, 72 stationary, 10, 19, 253 stopped just before T, 159, 292,

542 stopped, 23, 28, 51 process (cont’d)

variation — of another, 68, 226 Wiener —, 9 product σ-algebra, 402 product of elementary integrals,

infinite, 12, 404

, 402, 413 product paving, 402, 432, 434 progressively measurable, 25, 28, 35, 37, 38, 40, 41, 65, 437, 440, 510, 526, 552

projection, dual previsible, 221 predictable, 439 well-measurable, 440

projective limit,

of elementary integrals, 402, 404,

447 of probabilities, 164

projective system, full, 402, 447 , 401

Prokhoroff, 425

∥ ∥_{p}-semivariation, 53

pseudometric, 374 pseudometrizable, 375 punctured d-space, 180, 257

Q

quasi-left-continuity, of a Lévy process, 258 of a Markov process, 352 , 232, 235, 239, 250, 265, 285, 292,

319, 350 quasinorm, 381 quasinormed, 381 R Rademacher functions, 457 Radon measure, 177, 184, 231, 257,

263, 355, 394, 398, 413, 418, 442, 465, 469

Radon–Nikodym derivative, 41, 151,

223, 407, 450, 539 random,

interval, 28 partition, 138 partition, refinement of a, 138 sheet, 20 time, 27, 118, 436 vector field, 272

random function, predictable, 175 , 172, 180

randomly autologous,

coupling coefficient, 288, 300 random measure,

canonical representation, 177 compensated, 231 driving a SDE, 296 factorization of, 187, 208 local martingale–, 231 martingale–, 180 quasi-left-continuous, 232 , 56, 109, 173, 188, 205, 235, 246,

251, 263, 296, 370 spatially bounded, 173, 296 stopped, 173 strict previsible, 231 strict, 183, 231, 232 vanishing at 0, 173 Wiener, 178, 219

random partition,

refinement of a, 62 random time, graph of, 28 random variable,

nearly zero, 35 , 22, 391 simple, 46, 58, 391 symmetric stable, 458 (RC-0), 50 rcll, lcrl, 24 reals,

extended, 364, 375 rectification,

time — of a SDE, 280, 287 time — of a semigroup, 469

recurrent, 41

reduce, a process to a property, 51 a stopping time to a subset, 31,

118 reduced stopping time, 31 refine,

a filter, 373, 428

a random partition, 62, 138 reflection through the origin, 268,

410, 414 regular,

filtration, 38, 437

, 35

stochastic representation, 352 regularization,

of a filtration, 38, 135 relatively compact, 260, 264, 366, 385, 387, 425, 426, 428, 447

remainder, 277, 388, 390

remove a negligible set from Ω, 165,

166, 304 representation,

canonical of an integrator, 67 of a filtered probability space, 14,

64, 316 representation of martingales, for Lévy processes, 261 on Wiener space, 218

resolvent, identity, 463, 506 , 352, 463, 505

Riemann–Lebesgue lemma, 410 right-continuous, filtration, 37 process, 23

24 Index

right-continuous (cont’d)

, 44 time transformation, 550 version of a filtration, 37 version of a process, 24, 168

ring of sets, 394 Runge–Kutta, 281, 282, 321, 322,

327

S σ-additive, 394

in p-mean, 90, 106 marginally, 174, 371

σ-additivity, 90 σ-algebra, 394

Baire, 391 Baire vs. Borel, 391 function measurable on a, 391 generated by a family of func-

tions, 391 generated by a property, 391

σ-algebras, product of, 402 σ-algebra,

universally complete, 407 saturation,

of a collection of pseudometrics,

374 scalæfication, of processes, 139, 300,

312, 315, 335 scalae: ladder, flight of steps, 139,

312 Schwartz, 195, 205 Schwartz space, 269, 270, 410 σ-continuity, 90, 370, 394, 395, 566 SDE, Stoch. Differential Equation,

1 self-adjoint, 454 self–adjoint, 505 self-confined, 369 semicontinuous, 207, 376, 382 semigroup,

conservative, 467 continuous, 463 semigroup (cont’d)

contraction, 463 convolution, 254 Feller convolution, 268, 466 Feller, 268, 465 Gaussian, 19, 466 natural domain of a, 467 of operators, 463 Poisson, 359 resolvent of a, 463 time-rectification of a, 469

semimartingale, 232 seminorm, 380 semivariation, 53, 92 separable, 367

topological space, 15, 373, 377 separates the points, 367, 375, 399,

412, 425, 441, 447, 566 sequential closure, 537 sequential closure or span, 392 sequentially closed, 17, 391, 510 set,

analytic, 432 ℙ-nearly empty, 60 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-measurable, 114 identified with indicator function,

364 integrable, 104

σ-field, 394 sheet,

Brownian, 20 random, 20 Wiener —, 20

shift, a process, 4, 162, 164 a random measure, 186

σ-algebra, of \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-measurable sets, 114 optional — O, 440 P of predictable sets, 115 O of well-measurable sets, 440 σ-finite,

class of functions or sets, 392, 395,

397, 398, 406, 409 mean, 105, 112 measure, 406, 409, 416, 449

simple, measurable function, 448 point processs, 183 random variable, 46, 58, 391

size of a linear map, 381 Skorohod, 21, 443

space, 391 Skorohod topology, 21, 167, 411,

443, 445 slew of integrators,

see vector of integrators, 9 solid,

functional, 34 , 36, 90, 94

solution, strong, of a SDE, 273, 291 weak, of a SDE, 331

space, analytic, 441 completely regular, 373, 376 Hausdorff, 373 locally compact, 374 measurable, 391 polish, 15 Skorohod, 391

span, sequential, 392 sparse, 69, 165, 225 sparse previsible set, 235, 265 spatially bounded random measure,

173, 296 spectral radius, 506 spectrum, 505 spectrum of a function algebra, 367 square bracket, 148, 150 square function,

continuous, 148 of a complex integrator, 152 previsible, 228 square function (cont’d)

, 94, 148 square integrable,

locally — martingale, 84, 213 martingale, 78, 163, 186, 262 process, 72

square variation, 148, 149

stability,

of solutions to SDE’s, 50, 273,

293, 297 under change of measure, 129

stable, 220 stable span, 220 stable symmetric law, 458 standard deviation, 419 standard Wiener process,

d-dimensional, 20, 56, 160, 167,

218 on a filtration, 24, 72, 79, 298 , 11, 16, 18, 19, 77, 153, 162, 250,

326 state-of-the-world, 3 stationary,

background noise, 10 stationary process, 10, 19, 253 step function, 43 step size, 280, 311, 317, 319, 321, 327 stochastic,

analysis, 22, 34, 47, 436, 443 basis (see filtration), 22 exponential, 159, 163, 167, 180,

185, 219 flow, 343 integral, 99, 134 integral, elementary, 47 integrand, elementary, 46 integrator, 43, 50, 62 interval, bounded, 28 interval, finite, 28 partition, 138, 140, 147, 152, 169,

300, 312, 318 representation of a semigroup, 351 representation, regular, 352 Stone–Weierstraß, 108, 366, 377, 393,

399, 441, 442 stopped,

just before T, 159, 292, 542

process, 23, 28, 51 stopping,

optional — theorem, 77, 521 stopping time,

accessible, 122

announce a, 118, 284, 333, 525 arbitrarily large, 51 elementary, 47, 61 examples of, 29, 119, 437, 438 of bounded graph, 69 past of a, 28 predictable, 118, 438 reduced, 31 , 27, 51 totally inaccessible, 122, 232, 235,

236, 258 Stratonovich equation, 320, 321, 326 Stratonovich integral, 169, 320, 326 strictly positive or negative, 363 strict past,

of a stopping time, 530 strict past of a stopping time, 120 strict random measure, 183, 231, 232 strong law of large numbers, 76, 216 strong lifting, 419 strongly perpendicular, 220 strong Markov property, 349, 352 strong solution, 273, 291 strong type, 453 subadditive, 33, 34, 53, 54, 130, 380 ℙ-submartingale, 73 submartingale, 73, 74 subprobability, 80, 408, 465 ℙ-supermartingale, 73 supermartingale, 73, 74, 78, 81, 85,

352, 356 support of a measure, 400 sure control, 292 surely controlled, 292 Suslin,

space or subset, 441

, 432 symmetric form, 305 symmetric stable,

random variable or law, 458 symmetrization, 305 Szarek, 457

T

tail filter, 373

Taylor method, 281, 282, 321, 322,

327 Taylor’s formula, 387 THE,

Daniell mean, 89, 109, 124 previsible controller, 238 time transformation, 239, 283, 296

thread, 401

threshold, 7, 140, 168, 280, 311, 314 tiered weak derivatives, 306 Tietze’s extension theorem, 568 tight,

measure, 21, 165, 399, 407, 425,

441 uniformly, 334, 425, 427 time, random, 27, 118, 436 time-rectification of a SDE, 287 time-rectification of a semigroup,

469 time shift operator, 359 time transformation, the, 239, 283, 296 , 283, 444, 531, 550

topological space, Lusin, 441 polish, 20, 440 separable, 15, 373, 377 Suslin, 441 , 373

topological vector space, 379

topology, generated by functions, 411 topology (cont’d)

generated from functions, 376 Hausdorff, 373 of a uniformity, 374 of confined uniform convergence,

50, 172, 252, 370 of uniform convergence on com-

pacta, 14, 263, 372, 380, 385, 411, 426, 467

Skorohod, 21

, 373

uniform — on E, 51 totally bounded, 376 totally finite variation,

measure of, 394

, 45 totally inaccessible stopping time,

122, 232, 235, 236, 258 trajectory, 23 transformation, predictable, 185 transition probabilities, 465 translate, 269 translation-invariant, 269 transparent, 162 trapezoidal rule, 168 triangle inequality, 374 trick, 465, 469 Tychonoff’s theorem, 374, 425, 428 type,

map of weak —, 453

(p, q) of a map, 461

U ultimately constant, 119 ultrafilter, 373, 574

\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-a.e.,

defined process, 97

convergence, 96 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-integrable, 99 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-measurable, 111

process, on a set, 110

set, 114 \begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}^{∗}-negligible, 95

\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−p}^{∗}-a.e., 96

\begin{aligned}⌈ \\ ⌈ & ⌉ \\ & ⌉ \end{aligned}_{Z−p}^{∗}-negligible, 96

uniform, convergence, 111 largely — convergence, 111 uniform convergence on compacta,

see topology of, 380 uniform ellipticity, 337 uniformity,

generated by functions, 375, 405 induced on a subset, 374 , 374

E-uniformity, 110, 375 uniformizable, 375 uniformly continuous,

largely, 405 map, 374 pseudometric, 374 , 374

uniformly differentiable, weakly l-times, 305 weakly, 299, 300, 390 uniformly p-integrable,

collection of functions, 449 uniformly integrable, martingale, 72, 77 , 75, 225, 449

uniqueness of weak solutions, 331 unital, 504 universal completeness of the regu-

larization, 38 universal completion, 22, 26, 407,

436 universal integral, 141, 331 universally complete, filtration, 437, 440 , 38

universally measurable, function, 22 set or function, 23, 351, 407, 437 universal solution of an endogenous

SDE, 317, 347, 348 up-and-down procedure, 88

28 Index

upcrossing, 59, 74 upcrossing argument, 60, 75 upper integral, 32, 87, 396 upper regularity, 124 upper semicontinuous, 107, 194, 207,

376, 382 usual conditions, 39, 168 usual enlargement, 39, 168

V

vanish at infinity, 366, 367, 465 variation,

bounded, 45 finite, 45 function of finite, 45 measure of finite, 394 of a measure, 45, 394 process of bounded, 67 process of finite, 67 process, 68, 226 square, 148, 149 totally finite, 45 , 395

vector field, random, 272 , 272, 311

vector lattice, 366, 395 vector measure, 49, 53, 90, 108, 172,

448 vector of integrators,

see integrators, vector of, 9 version,

left-continuous, of a process, 24 right-continuous, of a process, 24 right-continuous, of a filtration,

37 W

Walsh basis, 196 weak,

derivative, 302, 390 higher order derivatives, 305 tiered derivatives, 306

weak convergence, of measures, 421 , 421

weak∗ topology, 263, 381 weakly differentiable,

l-times, 305 , 278, 390

weak solution, 331 weak topology, 381, 411 weak type, 453 well-measurable,

σ-algebra O, 440 process, 217, 440, 539

Wiener, integral, 5 measure, 16, 20, 426 random measure, 178, 219 sheet, 20 space, 16, 58 , 5

Wiener process, as integrator, 79, 220 canonical, 16, 58 characteristic function, 161 d-dimensional, 20, 218 Lévy’s characterization, 19, 160 on a filtration, 24, 72, 79, 298 square bracket, 153 standard d-dimensional, 20, 218 standard, 11, 16, 18, 19, 41, 77,

153, 162, 250, 326 , 8, 9, 10, 11, 17, 89, 149, 161, 162,

169, 251, 426 with covariance, 161, 258 Z Zero-One Law, 41, 256, 352, 358 Z-measurable, 129 Z-negligible, 129 Z−p-a.e., 96, 123

convergence, 96 Z-envelope,

predictable, 129 Z−p-envelope,

predictable, 135 Z−p-integrable, 99, 123 ζ−p-integrable, 175 Z−p-measurable, 111, 123 Z−p-negligible, 96