A characterization of stopping times
The authors give two characterizations of stopping times via martingales and Markov processes. The first result is the following: Let \(({\mathcal F}_ t)\) be a filtration satisfying the usual hypotheses and \(R\) be an \({\mathcal F}_ \infty\) random variable, then \(R\) is an \(({\mathcal F}_ t)\) stopping time if and only if \(E(H\mid {\mathcal F}_ R)= H_ R\) on \(\{R< \infty\}\), for all \(H\), where \(H\) is a bounded and \({\mathcal F}_ \infty\)- measurable r.v., \((H_ t)\) is a right continuous and left limited version of \(E(H\mid {\mathcal F}_ t)\) and \({\mathcal F}_ R= \sigma(Y_ R)\); \(Y\) a bounded, optional process. Replacing \({\mathcal F}_ R\) by \({\mathcal F}_{R-}\) and \(H_ R\) by \(H_{R-}\) we obtain the characterization of previsible stopping times. If \(X\) is a right continuous process having the strong Markov property, then \(R\) is a stopping time if and only if, given \({\mathcal F}_ R\), the conditional distribution of \(X_{R+.}\) is \(Z_ R\) where \(Z_ t\) is an optional version of the conditional distribution of \(X_{t+.}\) given \({\mathcal F}_ t\).
characterizations of stopping times via martingales
characterization of previsible stopping times
scientific article; zbMATH DE number 731995