Finite-time optimal control of a process leaving an interval

• Published in: Journal of applied probability 1996-09, Vol.33 (3), p.714-728
• Main Authors: Mcbeth, Douglas W., Weerasinghe, Ananda P. N.
• Format: Article
• Language: English
• ISSN: 0021-9002; 1475-6072
• DOI: 10.1017/S0021900200100154

Consider the optimal control problem of leaving an interval ( – a, a ) in a limited playing time. In the discrete-time problem, a is a positive integer and the player's position is given by a simple random walk on the integers with initial position x. At each time instant, the player chooses a coin from a control set where the probability of returning heads depends on the current position and the remaining amount of playing time, and the player is betting a unit value on the toss of the coin: heads returning +1 and tails − 1. We discuss the optimal strategy for this discrete-time game. In the continuous-time problem the player chooses infinitesimal mean and infinitesimal variance parameters from a control set which may depend upon the player's position. The problem is to find optimal mean and variance parameters that maximize the probability of leaving the interval [ — a, a ] within a finite time T > 0.