Dynkin's formula

In

.

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

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

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

Choose an

. Therefore, by Dynkin's formula,

Hence, for any j,

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

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