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.
be a Feller process with infinitesimal generator
For a point
denote the law of
given initial datum
denote expectation with respect to
, and any stopping time
, Dynkin's formula holds:[1] Let
-valued Itô diffusion solving the stochastic differential equation The infinitesimal generator
is defined by its action on compactly-supported
(twice differentiable with continuous second derivative) functions
is a Feller process, Dynkin's formula holds.
τ
is the first exit time of a bounded set
[ τ ] < + ∞
, then Dynkin's formula holds for all
, without the assumption of compact support.
[4] Dynkin's formula can be used to find the expected first exit time
τ
of a Brownian motion
from the closed ball
starts at a point
[5] Fix an integer j.
The strategy is to apply Dynkin's formula with
τ =
The generator of Brownian motion is
denotes the Laplacian operator.
Therefore, by Dynkin's formula, Hence, for any