Dynkin's formula

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 is named after the Russian mathematician Eugene Dynkin.

tatement of the theorem

Let "X" be the R^"n"-valued Itō diffusion solving the stochastic differential equation

:$mathrm\{d\}\; X\_\{t\}\; =\; b(X\_\{t\})\; ,\; mathrm\{d\}\; t\; +\; sigma\; (X\_\{t\})\; ,\; mathrm\{d\}\; B\_\{t\}.$

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 compactly-supported "C"² (twice differentiable with continuous second derivative) functions "f" : R^"n" → R as

:$A\; f\; (x)\; =\; lim\_\{t\; downarrow\; 0\}\; frac\{mathbf\{E\}^\{x\}\; [f(X\_\{t\})]\; -\; f(x)\}\{t\}$

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:

:$mathbf\{E\}^\{x\}\; [f(X\_\{\; au\})]\; =\; f(x)\; +\; mathbf\{E\}^\{x\}\; left\; [\; int\_\{0\}^\{\; au\}\; A\; f\; (X\_\{s\})\; ,\; mathrm\{d\}\; s\; ight]\; .$

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

:$K\; =\; K\_\{R\}\; =\; \{\; x\; in\; mathbf\{R\}^\{n\}\; |\; ,\; |\; x\; |\; leq\; R\; \},$

which, when "B" starts at a point "a" in the interior of "K", is given by

:

Choose an integer "k". The strategy is to apply Dynkin's formula with "X" = "B", "τ" = "σ"_"k" = min("k", "τ"_"K"), and a compactly-supported "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,

:::$=\; f(a)\; +\; mathbf\{E\}^\{a\}\; left\; [\; int\_\{0\}^\{sigma\_\{k\; frac1\{2\}\; Delta\; f\; (B\_\{s\})\; ,\; mathrm\{d\}\; s\; ight]$::$=\; |\; a\; |^\{2\}\; +\; mathbf\{E\}^\{a\}\; left\; [\; int\_\{0\}^\{sigma\_\{k\; n\; ,\; mathrm\{d\}\; s\; ight]$::$=\; |\; a\; |^\{2\}\; +\; n\; mathbf\{E\}^\{a\}\; [sigma\_\{k\}]\; .$

Hence, for any "k",

:

Now let "k" → +∞ to conclude that "τ"_"K" = lim_{"k"→+∞}"σ"_"k" < +∞ almost surely and

:

as claimed.

References

* cite book
last = Dynkin
first = Eugene B.
authorlink= Eugene Dynkin
coauthors = trans. J. Fabius, V. Greenberg, A. Maitra, G. Majone
title = Markov processes. Vols. I, II
series = Die Grundlehren der Mathematischen Wissenschaften, Bände 121
publisher = Academic Press Inc.
location = New York
year = 1965 (See Vol. I, p. 133)
* cite book
last = Øksendal
first = Bernt K.
authorlink = Bernt Øksendal
title = Stochastic Differential Equations: An Introduction with Applications
edition = Sixth edition
publisher=Springer
location = Berlin
year = 2003
id = ISBN 3-540-04758-1 (See Section 7.4)