Bernstein function and Martingale Problem: Difference between pages

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(Classical Martingale Problem) Given a (local) linear operator $\mathcal{L}: C^2(\mathbb{R}^n) \to C(\mathbb{R}^n)$ Martingale Problem for $\mathcal{L}$ consists in finding for each $x_0 \in \mathbb{R}^d$ a probability measure $\mathbb{P}^{x_0}$ over the space of all continuous functions $X: [0,+\infty) \to \mathbb{R}^d$ such that

\begin{equation*} \mathbb{P}^{x_0}\left ( X(0)=x_0 \right ) = 1 \end{equation*}

and whenever $f \in C^2(\mathbb{R}^2)$ we have that

\begin{equation*} f(X(t))-f(X(0))-\int_0^t \mathcal{L} f(X(s)) \;ds \end{equation*}

is a Local Martingale under $\mathbb{P}^{x_0}$

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-A continuous function $f : [0, \infty) \to [0, \infty)$ is said to be a Bernstein function if $(-1)^k f^{(k)}(x) \le 0$ for $x > 0$ and $k = 1, 2, ...$<ref name="SSV"/>
+{{stub}}
-==Relation to complete monotonicity==
+(Classical Martingale Problem) Given a (local) linear operator $\mathcal{L}: C^2(\mathbb{R}^n) \to C(\mathbb{R}^n)$ Martingale Problem for $\mathcal{L}$ consists in finding for each $x_0 \in \mathbb{R}^d$ a probability measure $\mathbb{P}^{x_0}$ over the space of all continuous functions $X: [0,+\infty) \to \mathbb{R}^d$  such that
-Clearly, $f$ is a Bernstein function if and only if it is nonnegative, and $f'$ is a [[completely monotone function]].
-==Representation==
+\begin{equation*}
-By Bernstein's theorem, $f$ is a Bernstein function if and only if:
+\mathbb{P}^{x_0}\left ( X(0)=x_0 \right ) = 1
-\[ f(z) = a z + b + \int_{(0, \infty)} (1 - e^{-t z}) \mu(\mathrm d t) \]
+\end{equation*}
-for some $a, b \ge 0$ and a Radon measure $\mu$ such that $\int_{(0, \infty)} \min(1, t) \mu(\mathrm d t) < \infty$.
-==Subordination==
+and whenever $f \in C^2(\mathbb{R}^2)$ we have that
-Bernstein functions are closely related to Bochner's [[subordination]] of semigroups. Namely, for a nonnegative definite self-adjoint operator $L$ and a Bernstein function $f$, the operator $-f(L)$ (defined by means of spectral theory) is the generator of some semigroup of operators which is subordinate to the semigroup $e^{-t L}$ generated by $-L$. Conversely, every generator of a semigroup subordinate to $e^{-t L}$ is equal to $-f(L)$ for some Bernstein function $f$.
-==Bernstein functions of the Laplacian==
+\begin{equation*}
-Bernstein functions of the Laplacian are translation invariant non-local operators in $\R^n$. More precisely, $A = f(-\Delta)$ for a Bernstein function $f$ if and only if
+f(X(t))-f(X(0))-\int_0^t \mathcal{L} f(X(s)) \;ds
-\[ -A u(x) = a \Delta u(x) + b u(x) + \int_{\R^n} (u(x + z) - u(x) - z \cdot \nabla u(x) \mathbf{1}_{|z| < 1}) k(z) \mathrm d z \]
+\end{equation*}
-for some $a, b \ge 0$ and $k(z)$ of the form
-\begin{align*}
-k(z) &= \int_0^\infty (4 \pi t)^{-n/2} e^{-|z|^2 / (4 t)} \mu(\mathrm d t) .
-\end{align*}
-==References==
+is a [[Martingale| Local Martingale]] under $\mathbb{P}^{x_0}$
-{{reflist|refs=
-<ref name="SSV">{{Citation | last1=Schilling | first1=R. | last2=Song | first2=R. | last3=Vondraček | first3=Z. | title=Bernstein functions. Theory and Applications | year=2010 | publisher=de Gruyter, Berlin | series=Studies in Mathematics | volume=37 | url=http://dx.doi.org/10.1515/9783110215311 | doi=10.1515/9783110215311}}</ref>
-}}
-{{stub}}

Bernstein function and Martingale Problem: Difference between pages

Revision as of 19:35, 19 November 2012

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