An Optimal Stopping Problem for Sums of Dichotomous Random Variables

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.