 
Summary: Applied Probability Trust (24 March 2004)
STOPPING THE MAXIMUM OF A CORRELATED RANDOM
WALK, WITH COST FOR OBSERVATION
PIETER ALLAART, University of North Texas
Abstract
Let (Sn)n0 be a correlated random walk on the integers, let M0
S0 be an arbitrary integer, and let Mn = max{M0, S1, . . . , Sn}.
An optimal stopping rule is derived for the sequence Mn  nc,
where c > 0 is a fixed cost. The optimal rule is shown to be of
threshold type: stop the first time that Mn Sn , where is a
certain nonnegative integer. An explicit expression for this optimal
threshold is given.
Keywords: Correlated random walk; momentum; stopping rule;
optimality principle; linear difference equation
AMS 2000 Subject Classification: Primary 60G40; 62L15
Secondary 60G50
1. Introduction
Consider a player owning a commodity whose price process exhibits
momentum in the following way: If the price goes up at stage n, it will
take another step up at stage n + 1 with probability p, or a step down
