Dynkin's formula

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 $\mathbf {P} ^{x}$ denote the law of $X$ given initial datum $X_{0}=x$, and let $\mathbf {E} ^{x}$ denote expectation with respect to $\mathbf {P} ^{x}$. Then for any function $f$ in the domain of $A$, and any stopping time $\tau$ with $\mathbf {E} [\tau ]<+\infty$, Dynkin's formula holds:

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

Example: Itô diffusions

Let $X$ be the $\mathbf {R} ^{n}$-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 $f:\mathbf {R} ^{n}\to \mathbf {R}$ as

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

or, equivalently,

$Af(x)=\sum _{i}b_{i}(x){\frac {\partial f}{\partial x_{i}}}(x)+{\frac {1}{2}}\sum _{i,j}{\big (}\sigma \sigma ^{\top }{\big )}_{i,j}(x){\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x).$

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

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 $K=\{x\in \mathbf {R} ^{n}:\,|x|\leq R\},$ which, when $B$ starts at a point $a$ in the interior of $K$, is given by

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

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 $\Delta /2$, where $\Delta$ denotes the Laplacian operator. Therefore, by Dynkin's formula,

${\begin{aligned}\mathbf {E} ^{a}\left[f{\big (}B_{\sigma _{j}}{\big )}\right]&=f(a)+\mathbf {E} ^{a}\left[\int _{0}^{\sigma _{j}}{\frac {1}{2}}\Delta f(B_{s})\,\mathrm {d} s\right]\\&=|a|^{2}+\mathbf {E} ^{a}\left[\int _{0}^{\sigma _{j}}n\,\mathrm {d} s\right]=|a|^{2}+n\mathbf {E} ^{a}[\sigma _{j}].\end{aligned}}$

Hence, for any $j$,

$\mathbf {E} ^{a}[\sigma _{j}]\leq {\frac {1}{n}}{\big (}R^{2}-|a|^{2}{\big )}.$

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

References

Sources

• Dynkin, Eugene B.; trans. J. Fabius; V. Greenberg; A. Maitra; G. Majone (1965). Markov processes. Vols. I, II. Die Grundlehren der Mathematischen Wissenschaften, Bände 121. New York: Academic Press Inc. (See Vol. I, p. 133)
• Kallenberg, Olav (2021). Foundations of Modern Probability (third ed.). Springer. ISBN 978-3-030-61870-4.
• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Section 7.4)