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 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:Kallenberg (2021), Lemma 17.21, p383.
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 (twice differentiable with continuous second derivative) functions asØksendal (2003), Definition 7.3.1, p124.
or, equivalently,Øksendal (2003), Theorem 7.3.3, p126.
Since this is a Feller process, Dynkin's formula holds.Øksendal (2003), Theorem 7.4.1, p127. 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.Øksendal (2003), Example 7.4.2, p127. 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.
References
Sources (See Vol. I, p. 133)
(See Section 7.4)