Martingales and Boundary Crossing Probabilities for Markov Processes

Abstract

Robbins and Siegmund have made use of the martingale $$\int^\infty_0 \exp(yW(t) - \frac{1}{2}ty^2) dF(y), t \geqq 0$$, to evaluate the probability that the Wiener process $W(t)$ would ever cross certain boundaries which are moving with time. By making use of martingales of the form $u(X(t), t)$, we apply the Robbins-Siegmund method to find boundary crossing probabilities for other Markov processes $X(t)$. The question of when $u(X(t), t)$ is a martingale is first studied. We generalize a result of Doob based on semigroups of type $\Gamma$, and we examine in particular the situations for stochastic integrals and processes with stationary independent increments.