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December, 1976 An Optimal Stopping Problem for Sums of Dichotomous Random Variables
H. Chernoff, A. J. Petkau
Ann. Probab. 4(6): 875-889 (December, 1976). DOI: 10.1214/aop/1176995933

Abstract

Let $Y_t$ be a stochastic process starting at $y$ which changes by i.i.d. dichotomous increments $X_t$ with mean 0 and variance 1. The cost of proceeding one step is one and the payoff is zero unless $n$ steps are taken and the final value $\hat{Y}$ of $Y_t$ is negative in which case the payoff is $\hat{Y}^2$. The optimal procedure consists of stopping as soon as $Y_t \geq \tilde{y}_m$ where $m$ is the number of steps left to be taken. The limit of $\tilde{y}_m$ as $m \rightarrow \infty$ is desired as a function of $p = P(X_t < 0)$. This limit $\tilde{y}$ is evaluated for $p$ rational and proved to be continuous in $p$. One can use $\tilde{y}$ to relate the solution of optimal stopping problems involving a Wiener process to those involving certain discrete-time discrete-process stopping problems. Thus $\tilde{y}$ is useful in calculating simple numerical approximations to solutions of various stopping problems.

Citation

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H. Chernoff. A. J. Petkau. "An Optimal Stopping Problem for Sums of Dichotomous Random Variables." Ann. Probab. 4 (6) 875 - 889, December, 1976. https://doi.org/10.1214/aop/1176995933

Information

Published: December, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0347.62064
MathSciNet: MR431559
Digital Object Identifier: 10.1214/aop/1176995933

Subjects:
Primary: 62L15
Secondary: 60G40

Rights: Copyright © 1976 Institute of Mathematical Statistics

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Vol.4 • No. 6 • December, 1976
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