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.
Publié le : 1976-12-14
Classification:
Backward induction,
difference equation,
free boundary problem,
optimal stopping,
Wiener process,
62L15,
60G40
@article{1176995933,
author = {Chernoff, H. and Petkau, A. J.},
title = {An Optimal Stopping Problem for Sums of Dichotomous Random Variables},
journal = {Ann. Probab.},
volume = {4},
number = {6},
year = {1976},
pages = { 875-889},
language = {en},
url = {http://dml.mathdoc.fr/item/1176995933}
}
Chernoff, H.; Petkau, A. J. An Optimal Stopping Problem for Sums of Dichotomous Random Variables. Ann. Probab., Tome 4 (1976) no. 6, pp. 875-889. http://gdmltest.u-ga.fr/item/1176995933/