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 infinitesimal generator of X, defined by its action on compactly-supportedC2 (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 setB ⊂ Rn with Ex[τ] < +∞, then Dynkin's formula holds for all C2 functions f, without the assumption of compact support.
Exampleedit
Dynkin's formula can be used to find the expected first exit time τK of Brownian motionB from the closed ball
which, when B starts at a point a in the interior of K, is given by
Choose an integerj. 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,
Hence, for any j,
Now let j → +∞ to conclude that τK = limj→+∞σj < +∞ almost surely and
as claimed.
Referencesedit
Dynkin, Eugene B.; trans. J. Fabius; V. Greenberg; A. Maitra; G. Majone (1965). Markov processes. Vols. I, II. Die Grundlehren der Mathematischen Wissenschaften, Bände 121. New York: Academic Press Inc. (See Vol. I, p. 133)
Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Section 7.4)