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Milds # Dynkin's formula

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.

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## 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

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

$Af(x)=\sum _{i}b_{i}(x){\frac {\partial f}{\partial x_{i}}}(x)+{\frac {1}{2}}\sum _{i,j}{\big (}\sigma \sigma ^{\top }{\big )}_{i,j}(x){\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}(x).$ Let τ be a stopping time with Ex[τ] < +∞, and let f be C2 with compact support. Then Dynkin's formula holds:

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

## 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

$\mathbf {E} ^{a}[\tau _{K}]={\frac {1}{n}}{\big (}R^{2}-|a|^{2}{\big )}.$ Choose an integer j. The strategy is to apply Dynkin's formula with X = B, τ = 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,

$\mathbf {E} ^{a}\left[f{\big (}B_{\sigma _{j}}{\big )}\right]$ $=f(a)+\mathbf {E} ^{a}\left[\int _{0}^{\sigma _{j}}{\frac {1}{2}}\Delta f(B_{s})\,\mathrm {d} s\right]$ $=|a|^{2}+\mathbf {E} ^{a}\left[\int _{0}^{\sigma _{j}}n\,\mathrm {d} s\right]$ $=|a|^{2}+n\mathbf {E} ^{a}[\sigma _{j}].$ Hence, for any j,

$\mathbf {E} ^{a}[\sigma _{j}]\leq {\frac {1}{n}}{\big (}R^{2}-|a|^{2}{\big )}.$ Now let j → +∞ to conclude that τK = limj→+∞σj < +∞ almost surely and

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

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