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A Generalization of Dynkin's Identity and Some Applications
Krishna B. Athreya and Thomas G. Kurtz
The Annals of Probability
Vol. 1, No. 4 (Aug., 1973), pp. 570579
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2959429
Page Count: 10
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Abstract
Let $X(t)$ be a right continuous temporally homogeneous Markov process, $T_t$ the corresponding semigroup and $A$ the weak infinitesimal generator. Let $g(t)$ be absolutely continuous and $\tau$ a stopping time satisfying $$E_x(\int^\tau_0 g(t) dt) < \infty \text{and} E_x(\int^\tau_0g'(t) dt) < \infty$$. Then for $f \in \mathscr{D}(A)$ with $f(X(t))$ right continuous the identity $$E_xg(\tau)f(X(\tau))  g(0)f(x) = E_x(\int^\tau_0 g'(s)f(X(s)) ds) + E_x(\int^\tau_0 g(s)Af(X(s)) ds)$$ is a simple generalization of Dynkin's identity $(g(t) \equiv 1)$. With further restrictions on $f$ and $\tau$ the following identity is obtained as a corollary: $$E_x(f(X(\tau))) = f(x) + \sum^{n1}_{k=1} \frac{(1)^{k1}}{k!} E_x(\tau^k A^k f(X(\tau))) \\ + \frac{(1)^{n1}}{(n1)!} E_x(\int^\tau_0 u^{n1}A^nf(X(u)) du)$$ 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.
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The Annals of Probability © 1973 Institute of Mathematical Statistics