Open Access
Translator Disclaimer
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


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.


Download Citation

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


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

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

Primary: 62L15
Secondary: 60G40

Keywords: backward induction , difference equation , free boundary problem , Optimal stopping , Wiener process

Rights: Copyright © 1976 Institute of Mathematical Statistics


Vol.4 • No. 6 • December, 1976
Back to Top