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 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

{\mathrm {d}}X_{{t}}=b(X_{{t}})\,{\mathrm {d}}t+\sigma (X_{{t}})\,{\mathrm {d}}B_{{t}}.\

For a point x ∈ Rⁿ, let P^x denote the law of X given initial datum X₀ = 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 C² (twice differentiable with continuous second derivative) functions f : Rⁿ → R as

Af(x)=\lim _{{t\downarrow 0}}{\frac {{\mathbf {E}}^{{x}}[f(X_{{t}})]-f(x)}{t}}\

or, equivalently,

Af(x)=\sum _{{i}}b_{{i}}(x){\frac {\partial f}{\partial x_{{i}}}}(x)+{\frac 1{2}}\sum _{{i,j}}{\big (}\sigma \sigma ^{{\top }}{\big )}_{{i,j}}(x){\frac {\partial ^{{2}}f}{\partial x_{{i}}\,\partial x_{{j}}}}(x).\

Let τ be a stopping time with E^x[τ] < +∞, and let f be C² with compact support. Then Dynkin's formula holds:

{\mathbf {E}}^{{x}}[f(X_{{\tau }})]=f(x)+{\mathbf {E}}^{{x}}\left[\int _{{0}}^{{\tau }}Af(X_{{s}})\,{\mathrm {d}}s\right].\

In fact, if τ is the first exit time for a bounded set B ⊂ Rⁿ with E^x[τ] < +∞, then Dynkin's formula holds for all C² 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

{\mathbf {E}}^{{a}}[\tau _{{K}}]={\frac 1{n}}{\big (}R^{{2}}-|a|^{{2}}{\big )}.

Choose an integer j. The strategy is to apply Dynkin's formula with X = B, τ = σ_j = min(j, τ_K), 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,

{\mathbf {E}}^{{a}}\left[f{\big (}B_{{\sigma _{{j}}}}{\big )}\right]

=f(a)+{\mathbf {E}}^{{a}}\left[\int _{{0}}^{{\sigma _{{j}}}}{\frac 1{2}}\Delta f(B_{{s}})\,{\mathrm {d}}s\right]

=|a|^{{2}}+{\mathbf {E}}^{{a}}\left[\int _{{0}}^{{\sigma _{{j}}}}n\,{\mathrm {d}}s\right]

=|a|^{{2}}+n{\mathbf {E}}^{{a}}[\sigma _{{j}}].

Hence, for any j,

{\mathbf {E}}^{{a}}[\sigma _{{j}}]\leq {\frac 1{n}}{\big (}R^{{2}}-|a|^{{2}}{\big )}.

Now let j → +∞ to conclude that τ_K = lim_j→+∞σ_j < +∞ almost surely and

{\mathbf {E}}^{{a}}[\tau _{{K}}]={\frac 1{n}}{\big (}R^{{2}}-|a|^{{2}}{\big )},

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 3-540-04758-1. (See Section 7.4)

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