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December, 1974 Martingales and Boundary Crossing Probabilities for Markov Processes
Tze Leung Lai
Ann. Probab. 2(6): 1152-1167 (December, 1974). DOI: 10.1214/aop/1176996503

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.

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Tze Leung Lai. "Martingales and Boundary Crossing Probabilities for Markov Processes." Ann. Probab. 2 (6) 1152 - 1167, December, 1974. https://doi.org/10.1214/aop/1176996503

Information

Published: December, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0326.60054
MathSciNet: MR436336
Digital Object Identifier: 10.1214/aop/1176996503

Keywords: 6060 , 6069 , 6245 , boundary crossing probabilities , infinitesimal generators , Martingales , processes with stationary independent increments , semigroups of type $\Gamma$ , stochastic integrals , Wald's equations

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 6 • December, 1974
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