A Generalization of Dynkin's Identity and Some Applications

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.