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 ${\displaystyle X}$ be a Feller process with infinitesimal generator ${\displaystyle A}$. For a point ${\displaystyle x}$ in the state-space of ${\displaystyle X}$, let ${\displaystyle \mathbf {P} ^{x}}$ denote the law of ${\displaystyle X}$ given initial datum ${\displaystyle X_{0}=x}$, and let ${\displaystyle \mathbf {E} ^{x}}$ denote expectation with respect to ${\displaystyle \mathbf {P} ^{x}}$. Then for any function ${\displaystyle f}$ in the domain of ${\displaystyle A}$, and any stopping time ${\displaystyle \tau }$ with ${\displaystyle \mathbf {E} [\tau ]<+\infty }$, Dynkin's formula holds:^[1]

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

Example: Itô diffusions

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

${\displaystyle \mathrm {d} X_{t}=b(X_{t})\,\mathrm {d} t+\sigma (X_{t})\,\mathrm {d} B_{t}.}$

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

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

or, equivalently,^[3]

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

Since this ${\displaystyle X}$ is a Feller process, Dynkin's formula holds.^[4] In fact, if ${\displaystyle \tau }$ is the first exit time of a bounded set ${\displaystyle B\subset \mathbf {R} ^{n}}$ with ${\displaystyle \mathbf {E} [\tau ]<+\infty }$, then Dynkin's formula holds for all ${\displaystyle C^{2}}$ functions ${\displaystyle f}$, without the assumption of compact support.^[4]

Application: Brownian motion exiting the ball

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

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

This is shown as follows.^[5] Fix an integer j. The strategy is to apply Dynkin's formula with ${\displaystyle X=B}$, ${\displaystyle \tau =\sigma _{j}=\min\{j,\tau _{K}\}}$, and a compactly-supported ${\displaystyle f\in C^{2}}$ with ${\displaystyle f(x)=|x|^{2}}$ on ${\displaystyle K}$. The generator of Brownian motion is ${\displaystyle \Delta /2}$, where ${\displaystyle \Delta }$ denotes the Laplacian operator. Therefore, by Dynkin's formula,

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

Hence, for any ${\displaystyle j}$,

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

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