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 function applied to a Feller process 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

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Let   be a Feller process with infinitesimal generator  . For a point   in the state-space of  , let   denote the law of   given initial datum  , and let   denote expectation with respect to  . Then for any function   in the domain of  , and any stopping time   with  , Dynkin's formula holds:[1]

 

Example: Itô diffusions

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Let   be the  -valued Itô diffusion solving the stochastic differential equation

 

The infinitesimal generator   of   is defined by its action on compactly-supported   (twice differentiable with continuous second derivative) functions   as[2]

 

or, equivalently,[3]

 

Since this   is a Feller process, Dynkin's formula holds.[4] In fact, if   is the first exit time of a bounded set   with  , then Dynkin's formula holds for all   functions  , without the assumption of compact support.[4]

Application: Brownian motion exiting the ball

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Dynkin's formula can be used to find the expected first exit time   of a Brownian motion   from the closed ball   which, when   starts at a point   in the interior of  , is given by

 

This is shown as follows.[5] Fix an integer j. The strategy is to apply Dynkin's formula with  ,  , and a compactly-supported   with   on  . The generator of Brownian motion is  , where   denotes the Laplacian operator. Therefore, by Dynkin's formula,

 

Hence, for any  ,

 

Now let   to conclude that   almost surely, and so   as claimed.

References

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  1. ^ Kallenberg (2021), Lemma 17.21, p383.
  2. ^ Øksendal (2003), Definition 7.3.1, p124.
  3. ^ Øksendal (2003), Theorem 7.3.3, p126.
  4. ^ a b Øksendal (2003), Theorem 7.4.1, p127.
  5. ^ Øksendal (2003), Example 7.4.2, p127.

Sources

  • 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)
  • Kallenberg, Olav (2021). Foundations of Modern Probability (third ed.). Springer. ISBN 978-3-030-61870-4.
  • Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN 3-540-04758-1. (See Section 7.4)