Dynkin's formula

In mathematics &mdash; specifically, in stochastic analysis &mdash; 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} A f (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


 * $$A f (x) = \lim_{t \downarrow 0} \frac{\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}$$

or, equivalently,


 * $$A f (x) = \sum_{i} b_{i} (x) \frac{\partial f}{\partial x_{i}} (x) + \frac1{2} \sum_{i, j} \big( \sigma \sigma^{\top} \big)_{i, j} (x) \frac{\partial^{2} f}{\partial x_{i}\, \partial x_{j}} (x).$$

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


 * $$\mathbf{E}^{a} [\tau_{K}] = \frac1{n} \big( R^{2} - | a |^{2} \big).$$

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{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$$,


 * $$\mathbf{E}^{a} [\sigma_{j}] \leq \frac1{n} \big( R^{2} - | a |^{2} \big).$$

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.