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 $$E_x(\int^\tau_0 |g(t)| dt) < \infty \text{and} E_x(\int^\tau_0|g'(t)| dt) < \infty$$. 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: $$E_x(f(X(\tau))) = f(x) + \sum^{n-1}_{k=1} \frac{(-1)^{k-1}}{k!} E_x(\tau^k A^k f(X(\tau))) \\ + \frac{(-1)^{n-1}}{(n-1)!} E_x(\int^\tau_0 u^{n-1}A^nf(X(u)) du)$$ 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.
Citation
Krishna B. Athreya. Thomas G. Kurtz. "A Generalization of Dynkin's Identity and Some Applications." Ann. Probab. 1 (4) 570 - 579, August, 1973. https://doi.org/10.1214/aop/1176996886
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