Statement of The Theorem
Let X be the Rn-valued Itō diffusion solving the stochastic differential equation
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
or, equivalently,
Let τ be a stopping time with Ex < +∞, and let f be C2 with compact support. Then Dynkin's formula holds:
In fact, if τ is the first exit time for a bounded set B ⊂ Rn with Ex < +∞, then Dynkin's formula holds for all C2 functions f, without the assumption of compact support.
Read more about this topic: Dynkin's Formula
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