The Annals of Probability

A Characterization of Stopping Times

Frank B. Knight and Bernard Maisonneuve

Full-text: Open access

Abstract

Let $R$ be a random time in $\mathscr{F}_\infty$, the terminal element of a filtration $\mathscr{F}_t$ satisfying the usual hypotheses. It is shown that if optimal sampling holds at $R$ for all bounded martingales, then $R$ is optional. If $\mathscr{F}_t$ is the natural pseudo-path filtration of a measurable process $X_t$, then $R$ is optional if (and only if) the conditional distribution of $X_{R + .}$ given $\mathscr{F}_R$ is $Z_R$, where $Z_t$ is an optional version of the conditional distribution of $X_{t +.}$ given $\mathscr{F}_t$.

Article information

Source
Ann. Probab., Volume 22, Number 3 (1994), 1600-1606.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176988615

Digital Object Identifier
doi:10.1214/aop/1176988615

Mathematical Reviews number (MathSciNet)
MR1303657

Zentralblatt MATH identifier
0816.60039

JSTOR
links.jstor.org

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60G05: Foundations of stochastic processes 60G07: General theory of processes 60G25: Prediction theory [See also 62M20] 60G44: Martingales with continuous parameter 60J25: Continuous-time Markov processes on general state spaces

Keywords
Stopping times optional times random times martingales Markov processes prediction process

Citation

Knight, Frank B.; Maisonneuve, Bernard. A Characterization of Stopping Times. Ann. Probab. 22 (1994), no. 3, 1600--1606. doi:10.1214/aop/1176988615. https://projecteuclid.org/euclid.aop/1176988615


Export citation