Dynkin's formula
In mathematics — specifically, in — Dynkin's formula is a theorem giving the expected value of any suitably smooth statistic of an Itō diffusion 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.
Let X be the Rn-valued Itō diffusion solving the
For a point x ∈ Rn, let Px denote the law of X given initial datum X0 = x, and let Ex denote expectation with respect to Px.
Let A be the of X, defined by its action on compactly-supported C2 (twice differentiable with continuous second derivative) functions f : Rn → R as
or, equivalently,
Let τ be a stopping time with Ex[τ] < +∞, and let f be C2 with compact support. Then Dynkin's formula holds:
In fact, if τ is the first exit time for a bounded set B ⊂ Rn with Ex[τ] < +∞, then Dynkin's formula holds for all C2 functions f, without the assumption of compact support.
Dynkin's formula can be used to find the expected first exit time τK of Brownian motion B from the closed ball
which, when B starts at a point a in the interior of K, is given by
Choose an integer j. The strategy is to apply Dynkin's formula with X = B, τ = σj = min(j, τK), and a compactly-supported C2f with f(x) = |x|2 on K. The generator of Brownian motion is Δ/2, where Δ denotes the Laplacian operator. Therefore, by Dynkin's formula,
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