Dynkin's formula

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In mathematics — specifically, in stochastic analysisDynkin'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[edit]

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.


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 j. The strategy is to apply Dynkin's formula with X = B, τ = σj = min(jτK), 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}} \frac1{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 \frac1{n} \big( R^{2} - | a |^{2} \big).

Now let j → +∞ to conclude that τK = limj→+∞σj < +∞ almost surely and

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

as claimed.


  • 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)