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 R^''n''-valued Itō diffusion solving the stochastic differential equation

:$\mathrm X_ = b(X_) \, \mathrm t + \sigma (X_) \, \mathrm B_.$

For a point ''x'' ∈ R^''n'', let P^''x'' denote the law of ''X'' given initial datum ''X''₀ = ''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 ''C''² (twice differentiable with continuous second derivative) functions ''f'' : R^''n'' → R as

:$A f (x) = \lim_ \frac$

or, equivalently,

:$A f (x) = \sum_ b_ (x) \frac (x) + \frac1 \sum_ \big( \sigma \sigma^ \big)_ (x) \frac (x).$

Let ''τ'' be a stopping time with E^''x'' 'τ''nbsp;< +∞, and let ''f'' be ''C''² with compact support. Then Dynkin's formula holds:

:\mathbf^ (X_)= f(x) + \mathbf^ \left \int_^ A f (X_) \, \mathrm s \right

In fact, if ''τ'' is the first exit time for a bounded set ''B'' ⊂ R^''n'' with E^''x'' 'τ''nbsp;< +∞, then Dynkin's formula holds for all ''C''² 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

$K = K_ = \,$

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

\mathbf^ tau_= \frac1 \big( R^ - , a , ^ \big).

Choose an integer ''j''. The strategy is to apply Dynkin's formula with ''X'' = ''B'', ''τ'' = ''σ''_''j'' = min(''j'', ''τ''_''K''), and a compactly-supported ''C''² ''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,

\begin
\mathbf^ \left f \big( B_ \big) \right&= f(a) + \mathbf^ \left \int_^ \frac1 \Delta f (B_) \, \mathrm s \right\\
&= , a , ^ + \mathbf^ \left \int_^ n \, \mathrm s \right\\
&= , a , ^ + n \mathbf^ sigma_
\end

Hence, for any ''j'',
\mathbf^ sigma_\leq \frac1 \big( R^ - , a , ^ \big).

Now let ''j'' → +∞ to conclude that ''τ''_''K'' = lim_''j''→+∞''σ''_''j'' < +∞ almost surely and

\mathbf^ tau_= \frac1 \big( R^ - , a , ^ \big),

as claimed.

References

* (See Vol. I, p. 133)
* {{cite book
, last = Øksendal
, first = Bernt K.
, authorlink = Bernt Øksendal
, title = Stochastic Differential Equations: An Introduction with Applications
, edition = Sixth
, publisher=Springer
, location = Berlin
, year = 2003
, isbn = 3-540-04758-1
(See Section 7.4)

Stochastic differential equations
Probability theorems