Optimal Stopping Variables for Stochastic Process with Independent Increments

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.