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

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

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

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

or, equivalently,

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

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

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

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

${\displaystyle K=K_{R}=\{x\in \mathbf {R} ^{n}|\,|x|\leq R\},}$

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

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

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,

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

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

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

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

Hence, for any j,

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

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

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

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)