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.
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Transcription
Statement of the theorem
Let X be the R^{n}valued Itō diffusion solving the stochastic differential equation
For a point x ∈ R^{n}, let P^{x} denote the law of X given initial datum X_{0} = x, and let E^{x} denote expectation with respect to P^{x}.
Let A be the infinitesimal generator of X, defined by its action on compactlysupported C^{2} (twice differentiable with continuous second derivative) functions f : R^{n} → R as
or, equivalently,
Let τ be a stopping time with E^{x}[τ] < +∞, and let f be C^{2} with compact support. Then Dynkin's formula holds:
In fact, if τ is the first exit time for a bounded set B ⊂ R^{n} with E^{x}[τ] < +∞, then Dynkin's formula holds for all C^{2} functions f, without the assumption of compact support.
Example
Dynkin's formula can be used to find the expected first exit time τ_{K} 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 = σ_{j} = min(j, τ_{K}), and a compactlysupported C^{2} 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 τ_{K} = lim_{j→+∞}σ_{j} < +∞ almost surely and
as claimed.
References
 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 3540047581. (See Section 7.4)