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.
Publié le : 1974-12-14
Classification:
6060,
6069,
6245,
Martingales,
boundary crossing probabilities,
semigroups of type $\Gamma$,
infinitesimal generators,
processes with stationary independent increments,
Wald's equations,
stochastic integrals
@article{1176996503,
author = {Lai, Tze Leung},
title = {Martingales and Boundary Crossing Probabilities for Markov Processes},
journal = {Ann. Probab.},
volume = {2},
number = {6},
year = {1974},
pages = { 1152-1167},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996503}
}
Lai, Tze Leung. Martingales and Boundary Crossing Probabilities for Markov Processes. Ann. Probab., Tome 2 (1974) no. 6, pp. 1152-1167. http://gdmltest.u-ga.fr/item/1176996503/