Dynkin's Formula - Statement of The Theorem

Statement of The Theorem

Let X be the Rn-valued Itō diffusion solving the stochastic differential equation

For a point xRn, 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 : RnR 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 BRn 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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