Dynkin's formula

Template:Short description In mathematics — specifically, in stochastic analysis — Dynkin's formula is a theorem giving the expected value of any suitably smooth function applied to a Feller process at a stopping time. It may be seen as a stochastic generalization of the (second) fundamental theorem of calculus. It is named after the Russian mathematician Eugene Dynkin.

Statement of the theorem

Let $X$ be a Feller process with infinitesimal generator $A$. For a point $x$ in the state-space of $X$, let ${𝐏}^x$ denote the law of $X$ given initial datum $X_0=x$, and let ${𝐄}^x$ denote expectation with respect to ${𝐏}^x$. Then for any function $f$ in the domain of $A$, and any stopping time $\tau$ with $𝐄[\tau ]<+\mathrm{\infty }$, Dynkin's formula holds:

${𝐄}^x[f(X_{\tau })]=f(x)+{𝐄}^x[\underset{0}{\overset{\tau }{\int }}Af(X_s)\mathrm{d}s].$

Example: Itô diffusions

Let $X$ be the ${𝐑}^n$-valued Itô diffusion solving the stochastic differential equation

$\mathrm{d}{X}_t=b(X_t)\mathrm{d}t+\sigma (X_t)\mathrm{d}{B}_t.$

The infinitesimal generator $A$ of $X$ is defined by its action on compactly-supported $C^2$ (twice differentiable with continuous second derivative) functions $f:{𝐑}^n\to 𝐑$ as

$Af(x)=\underset{t\downarrow 0}{\lim }\frac{{𝐄}^x[f(X_t)]-f(x)}{t}$

or, equivalently,

$Af(x)=\sum _i{b}_i(x)\frac{\partial f}{\partial x_i}(x)+\frac{1}{2}\sum _{i,j}(\sigma {\sigma }^{\mathrm{\top }}{)}_{i,j}(x)\frac{{\partial }^{2}f}{\partial x_i\partial x_j}(x).$

Since this $X$ is a Feller process, Dynkin's formula holds. In fact, if $\tau$ is the first exit time of a bounded set $B\subset {𝐑}^n$ with $𝐄[\tau ]<+\mathrm{\infty }$, then Dynkin's formula holds for all $C^2$ functions $f$, without the assumption of compact support.Template:R

Application: Brownian motion exiting the ball

Dynkin's formula can be used to find the expected first exit time ${\tau }_K$ of a Brownian motion $B$ from the closed ball $K=\{x\in {𝐑}^n:|x|\le R\},$ which, when $B$ starts at a point $a$ in the interior of $K$, is given by

${𝐄}^a[{\tau }_K]=\frac{1}{n}(R^2-|a{|}^2).$

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

$\begin{array}{cc}{𝐄}^a\left[f\right(B_{{\sigma }_j}\left)\right]& =f(a)+{𝐄}^a[\underset{0}{\overset{{\sigma }_j}{\int }}\frac{1}{2}\mathrm{\Delta }f(B_s)\mathrm{d}s]\\ & =|a{|}^2+{𝐄}^a\left[\underset{0}{\overset{{\sigma }_j}{\int }}n\mathrm{d}s\right]=|a{|}^2+n{𝐄}^a[{\sigma }_j].\end{array}$

Hence, for any $j$,

${𝐄}^a[{\sigma }_j]\le \frac{1}{n}(R^2-|a{|}^2).$

Now let $j\to +\mathrm{\infty }$ to conclude that ${\tau }_K=\underset{j\to +\mathrm{\infty }}{\lim }{\sigma }_j<+\mathrm{\infty }$ almost surely, and so ${𝐄}^a[{\tau }_K]=(R^2-|a{|}^2)/n$ as claimed.

References

Sources

• Template:Cite book (See Vol. I, p. 133)
• Template:Cite book
• Template:Cite book (See Section 7.4)