Appl Math Optim 45:283–324 (2002)
2001 Springer-Verlag New York Inc.
Brownian Optimal Stopping and Random Walks
Equipe d’Analyse et de Math´ematiques Appliqu´ees, Universit´e de Marne-la-Vall´ee,
5 Boulevard Descartes, Cit´e Descartes, Champs-sur-Marne,
77 454 Marne-la-Vall´ee Cedex 2, France
Abstract. One way to compute the value function of an optimal stopping problem
along Brownian paths consists of approximating Brownian motion by a random
walk. We derive error estimates for this type of approximation under various as-
sumptions on the distribution of the approximating random walk.
Key Words. Optimal stopping, Brownian motion, Random walk approximation,
AMS Classiﬁcation. 60G40, 90A09.
This paper deals with the approximation of optimal stopping problems of the following
type. Let B = (B
be a standard Brownian motion on a bounded interval [0, T ],
and let f be a bounded continuous function on the real line. Consider the number
P = sup
f (µτ + B
where r and µ are real constants and T
denotes the set of all stopping times of the
natural ﬁltration of B, with values in the interval [0, T]. In applications, the constant r is
a discount rate and, in general, is positive, but this will not be needed in our discussion.
One way to compute P numerically is to approximate B by a random walk and
apply dynamic programming. To be more precise, assume (X
is a sequence of i.i.d.
real random variables satisfying EX
= 1 and EX
= 0, and deﬁne, for any positive