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.
Publié le : 1974-04-14
Classification:
Optimal stopping variables (rules),
stochastic processes with independent,
statistically homogeneous,
zero mean,
bounded variance increments,
right-continuous paths,
65L15,
60G40,
60J30
@article{1176996710,
author = {Walker, LeRoy H.},
title = {Optimal Stopping Variables for Stochastic Process with Independent Increments},
journal = {Ann. Probab.},
volume = {2},
number = {6},
year = {1974},
pages = { 309-316},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996710}
}
Walker, LeRoy H. Optimal Stopping Variables for Stochastic Process with Independent Increments. Ann. Probab., Tome 2 (1974) no. 6, pp. 309-316. http://gdmltest.u-ga.fr/item/1176996710/