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 fundamental theorem of calculus. It is named after the Russian mathematician Eugene Dynkin.

Statement of the theorem

Let be a Feller process with infinitesimal generator.
For a point in the state-space of, let denote the law of given initial datum, and let denote expectation with respect to.
Then for any function in the domain of, and any stopping time with, Dynkin's formula holds:

Example: Itô diffusions

Let be the -valued Itô diffusion solving the stochastic differential equation
The infinitesimal generator of is defined by its action on compactly-supported functions as
or, equivalently,
Since this is a Feller process, Dynkin's formula holds.
In fact, if is the first exit time of a bounded set with, then Dynkin's formula holds for all functions, 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 of a Brownian motion from the closed ball
which, when starts at a point in the interior of, is given by
This is shown as follows. Fix an integer j. The strategy is to apply Dynkin's formula with,, and a compactly-supported with on. The generator of Brownian motion is, where denotes the Laplacian operator. Therefore, by Dynkin's formula,
Hence, for any,
Now let to conclude that almost surely, and so
as claimed.