Dynkin's formula

Let $X$ be a Feller process with infinitesimal generator $A$.

For a point $x$ in the state-space of $X$, let $\backslash mathbf\; P^x$ denote the law of $X$ given initial datum $X\_0=x$, and let $\backslash mathbf\; E^x$ denote expectation with respect to $\backslash mathbf\; P^x$.

Then for any function $f$ in the domain of $A$, and any stopping time $\backslash tau$ with , Dynkin's formula holds:Kallenberg (2021), Lemma 17.21, p383.

:$$

\mathbf{E}^{x} [f(X_{\tau})] = f(x) + \mathbf{E}^{x} \left[ \int_{0}^{\tau} A f (X_{s}) \, \mathrm{d} s \right].

Let $X$ be the $\backslash mathbf\; R^n$-valued Itô diffusion solving the stochastic differential equation

:$\backslash mathrm\{d\}\; X\_\{t\}\; =\; b(X\_\{t\})\; \backslash ,\; \backslash mathrm\{d\}\; t\; +\; \backslash sigma\; (X\_\{t\})\; \backslash ,\; \backslash mathrm\{d\}\; B\_\{t\}.$

The infinitesimal generator $A$ of $X$ is defined by its action on compactly-supported $C^2$ (twice differentiable with continuous second derivative) functions $f:\backslash mathbf\; R^n\; \backslash to\; \backslash mathbf\; R$ asØksendal (2003), Definition 7.3.1, p124.

:$A\; f\; (x)\; =\; \backslash lim\_\{t\; \backslash downarrow\; 0\}\; \backslash frac\{\backslash mathbf\{E\}^\{x\}\; [f(X\_\{t\})]\; -\; f(x)\}\{t\}$

or, equivalently,Øksendal (2003), Theorem 7.3.3, p126.

:$A\; f\; (x)\; =\; \backslash sum\_\{i\}\; b\_\{i\}\; (x)\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\_\{i\}\}\; (x)\; +\; \backslash frac1\{2\}\; \backslash sum\_\{i,\; j\}\; \backslash big(\; \backslash sigma\; \backslash sigma^\{\backslash top\}\; \backslash big)\_\{i,\; j\}\; (x)\; \backslash frac\{\backslash partial^\{2\}\; f\}\{\backslash partial\; x\_\{i\}\backslash ,\; \backslash partial\; x\_\{j\}\}\; (x).$

Since this $X$ is a Feller process, Dynkin's formula holds.Øksendal (2003), Theorem 7.4.1, p127.

In fact, if $\backslash tau$ is the first exit time of a bounded set $B\backslash subset\backslash mathbf\; R^n$ with , then Dynkin's formula holds for all $C^2$ functions $f$, without the assumption of compact support.{{r|oksendal}}

Dynkin's formula can be used to find the expected first exit time $\backslash tau\_K$ of a Brownian motion $B$ from the closed ball

$K=\; \backslash \{\; x\; \backslash in\; \backslash mathbf\{R\}^\{n\}\; :\; \backslash ,\; |\; x\; |\; \backslash leq\; R\; \backslash \},$

which, when $B$ starts at a point $a$ in the interior of $K$, is given by

:$\backslash mathbf\{E\}^\{a\}\; [\backslash tau\_\{K\}]\; =\; \backslash frac1\{n\}\; \backslash big(\; R^\{2\}\; -\; |\; a\; |^\{2\}\; \backslash big).$

This is shown as follows.Øksendal (2003), Example 7.4.2, p127. Fix an integer j. The strategy is to apply Dynkin's formula with $X=B$, $\backslash tau=\backslash sigma\_j=\backslash min\backslash \{j,\backslash tau\_K\backslash \}$, and a compactly-supported $f\backslash in\; C^2$ with $f(x)=|x|^2$ on $K$. The generator of Brownian motion is $\backslash Delta/2$, where $\backslash Delta$ denotes the Laplacian operator. Therefore, by Dynkin's formula,

:$\backslash begin\{align\}$

\mathbf{E}^{a} \left[ f \big( B_{\sigma_{j}} \big) \right]

&= f(a) + \mathbf{E}^{a} \left[ \int_{0}^{\sigma_{j}} \frac1{2} \Delta f (B_{s}) \, \mathrm{d} s \right] \\

&= | a |^{2} + \mathbf{E}^{a} \left[ \int_{0}^{\sigma_{j}} n \, \mathrm{d} s \right]

= | a |^{2} + n \mathbf{E}^{a} [\sigma_{j}].

\end{align}

Hence, for any $j$,

:$\backslash mathbf\{E\}^\{a\}\; [\backslash sigma\_\{j\}]\; \backslash leq\; \backslash frac1\{n\}\; \backslash big(\; R^\{2\}\; -\; |\; a\; |^\{2\}\; \backslash big).$

Now let $j\backslash to+\backslash infty$ to conclude that almost surely, and so

$\backslash mathbf\{E\}^\{a\}\; [\backslash tau\_\{K\}]\; =(\; R^\{2\}\; -\; |\; a\; |^\{2\})/n$

as claimed.

{{reflist}}

Sources

• {{cite book

| last = Dynkin

| first = Eugene B.

|authorlink= Eugene Dynkin

|author2=trans. J. Fabius |author3=V. Greenberg |author4=A. Maitra |author5=G. Majone

| title = Markov processes. Vols. I, II

| series = Die Grundlehren der Mathematischen Wissenschaften, Bände 121

|publisher = Academic Press Inc.

| location = New York

| year = 1965

}} (See Vol. I, p. 133)

• {{cite book

| last = Kallenberg

| first = Olav

| authorlink = Olav Kallenberg

| title = Foundations of Modern Probability

| edition = third

| publisher = Springer

| year = 2021

| isbn = 978-3-030-61870-4

}}

• {{cite book

| last = Øksendal

| first = Bernt K.

| authorlink = Bernt Øksendal

| title = Stochastic Differential Equations: An Introduction with Applications

| edition = Sixth

| publisher=Springer

| location = Berlin

| year = 2003

| isbn = 3-540-04758-1

}} (See Section 7.4)

Category:Stochastic differential equations

Category:Theorems about stochastic processes