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April, 1974 Optimal Stopping Variables for Stochastic Process with Independent Increments
LeRoy H. Walker
Ann. Probab. 2(2): 309-316 (April, 1974). DOI: 10.1214/aop/1176996710

Abstract

Let $\{W(t): t$ a nonnegative real number$\}$ denote a stochastic process with right-continuous sample paths with probability one, independent increments which are statistically homogeneous, $E\{W(t)\} = 0$, and $E\{(W(t) - W(s))^2\} = \sigma|t - s|$ for some constant $\sigma$; let $T$ denote the set of stopping variables with respect to $W$; and let $c$ denote a non-increasing, right-continuous, square-integrable function on the nonnegative real line. Then $E\{\sup_{t\geqq 0} c(t)|W(t)|\}$ is shown to be finite which insures that $\sup_{\tau\in T} E\{c(\tau)W(\tau)\}$ is finite. Also, $\varepsilon$-optimal stopping variables are shown to exist with stopping points occurring only in discrete subsets of the nonnegative real line. These optimal stopping variables require observation of the process $W$ only at the possible stopping points.

Citation

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LeRoy H. Walker. "Optimal Stopping Variables for Stochastic Process with Independent Increments." Ann. Probab. 2 (2) 309 - 316, April, 1974. https://doi.org/10.1214/aop/1176996710

Information

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

zbMATH: 0282.60028
MathSciNet: MR397866
Digital Object Identifier: 10.1214/aop/1176996710

Subjects:
Primary: 65L15
Secondary: 60G40 , 60J30

Keywords: bounded variance increments , Optimal stopping variables (rules) , right-continuous paths , statistically homogeneous , stochastic processes with independent , zero mean

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 2 • April, 1974
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