In mathematics — specifically, in stochastic analysis — 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.
Statement of the theorem
Let X be the Rn-valued Itō diffusion solving the stochastic differential equation
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 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.
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 C2 f 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
- 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 edition ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Section 7.4)