Dynkin's formula

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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.

[edit] 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 Px denote the law of X given initial datum X0 = x, and let Ex denote expectation with respect to Px.

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 Ex[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 Ex[τ] < +∞, then Dynkin's formula holds for all C2 functions f, without the assumption of compact support.

[edit] Example

Dynkin's formula can be used to find the expected first exit time τK 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 = σk = min(k, τK), 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 τK = limk→+∞σk < +∞ almost surely and

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

as claimed.

[edit] 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, Berlin: Springer. ISBN 3-540-04758-1.  (See Section 7.4)