Dynkin's formula

Inmathematics— specifically, instochastic analysis—Dynkin's formulais a theorem giving theexpected valueof any suitably smooth function applied to aFeller processat astopping time.It may be seen as a stochastic generalization of the (second)fundamental theorem of calculus.It is named after theRussianmathematicianEugene Dynkin.

Statement of the theorem

Let$X$be a Feller process withinfinitesimal 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 anystopping time$\tau$with$\mathbf {E} [\tau ]<+\infty$,Dynkin's formulaholds:

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

Example: Itô diffusions

Let$X$be the$\mathbf {R} ^{n}$-valuedItô diffusionsolving thestochastic 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 oncompactly-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 abounded 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 aBrownian motion$B$from theclosed ball $K=\{x\in \mathbf {R} ^{n}:\,|x|\leq R\},$ which, when$B$starts at a point$a$in theinteriorof$K$,is given by

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

This is shown as follows.Fix anintegerj.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 theLaplacian 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.ISBN978-3-030-61870-4.
• Øksendal, Bernt K.(2003).Stochastic Differential Equations: An Introduction with Applications(Sixth ed.). Berlin: Springer.ISBN3-540-04758-1.(See Section 7.4)