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
Famous quotes containing the words statement of the, statement of, statement and/or theorem:
“It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
—John Dewey (1859–1952)
“The honor my country shall never be stained by an apology from me for the statement of truth and the performance of duty; nor can I give any explanation of my official acts except such as is due to integrity and justice and consistent with the principles on which our institutions have been framed.”
—Andrew Jackson (1767–1845)
“It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
—John Dewey (1859–1952)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)