Dynkin's formula

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In mathematics — specifically, in stochastic analysisDynkin'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[edit]

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 A be the infinitesimal generator 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.

Example[edit]

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(jτK), 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

as claimed.

References[edit]

  • 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)