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Generalization of Dynkin's Identity and Some Applications

Athreya, KB and Kurtz, TG (1973) Generalization of Dynkin's Identity and Some Applications. In: Annals of Probability, 1 (4). pp. 570-579.

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Abstract

Let X(t) be a right continuous temporally homogeneous Markov pro- cess, Tt the corresponding semigroup and A the weak infinitesimal genera- tor. Let g(t) be absolutely continuous and r a stopping time satisfying E.( S f I g(t) I dt) < oo and E.( f " I g'(t) I dt) < oo Then for f e 9iJ(A) with f(X(t)) right continuous the identity Exg(r)f(X(z)) - g(O)f(x) = E( 5 " g'(s)f(X(s)) ds) + E.( 5 " g(s)Af(X(s)) ds) is a simple generalization of Dynkin's identity (g(t) 1). With further restrictions on f and r the following identity is obtained as a corollary: Ex(f(X(z))) = f(x) + k! Ex~rkAkf(X(z))) + n-1E + (n ) )!.E,(so un-1Anf(X(u)) du). These identities are applied to processes with stationary independent increments to obtain a number of new and known results relating the moments of stopping times to the moments of the stopped processes.

Item Type: Editorials/Short Communications
Additional Information: Copyright of this article belongs to Institute of Mathematical Statistics.
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 14 Jul 2010 04:00
Last Modified: 19 Sep 2010 06:10
URI: http://eprints.iisc.ernet.in/id/eprint/28706

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