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

$A f (x) = \lim_{t \downarrow 0} \frac{\mathbf{E}^{x} [f(X_{t})] - f(x)}{t}\$

or, equivalently,

$A f (x) = \sum_{i} b_{i} (x) \frac{\partial f}{\partial x_{i}} (x) + \frac1{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} A f (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

$\mathbf{E}^{a} [\tau_{K}] = \frac1{n} \big( R^{2} - | a |^{2} \big).$

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 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,

$\mathbf{E}^{a} \left[ f \big( B_{\sigma_{k}} \big) \right]$

$= f(a) + \mathbf{E}^{a} \left[ \int_{0}^{\sigma_{k}} \frac1{2} \Delta f (B_{s}) \, \mathrm{d} s \right]$

$= | a |^{2} + \mathbf{E}^{a} \left[ \int_{0}^{\sigma_{k}} n \, \mathrm{d} s \right]$

$= | a |^{2} + n \mathbf{E}^{a} [\sigma_{k}].$

Hence, for any k,

$\mathbf{E}^{a} [\sigma_{k}] \leq \frac1{n} \big( R^{2} - | a |^{2} \big).$

Now let k → +∞ to conclude that τ_K = lim_k→+∞σ_k < +∞ almost surely and

$\mathbf{E}^{a} [\tau_{K}] = \frac1{n} \big( R^{2} - | a |^{2} \big),$

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 edition ed.). Berlin: Springer. ISBN 3-540-04758-1.  (See Section 7.4)