Maximizing the Mean Exit Time of a Brownian Motion from an Interval

Mario Lefebvre

doi:10.1155/2011/296259

Abstract

Let X(t) be a controlled one-dimensional standard Brownian motion starting from x∈(−d,d). The problem of optimally controlling X(t) until |X(t)|=d for the first time is solved explicitly in a particular case. The maximal value that the instantaneous reward given for survival in (−d,d) can take is determined.

Highlights

Consider the one-dimensional controlled standard Brownian motion process {X t, t ≥ 0} defined by the stochastic differential equation dX t b0 XtkuXt dt dB t, 1.1 where u is the control variable, b0 > 0, k ∈ {0, 1, . . .} and {B t, t ≥ 0} is a standard Brownian motion
It follows that the optimal control is given by u∗ x − b0 x F x
Because the optimizer wants X t to remain in the interval −d, d as long as possible and because u X t is multiplied by b0X t with b0 > 0 in 1.1, we can state that the optimal control u∗ x should always be negative when x / 0

Summary

Introduction

Consider the one-dimensional controlled standard Brownian motion process {X t , t ≥ 0} defined by the stochastic differential equation dX t b0 XtkuXt dt dB t , 1.1 where u is the control variable, b0 > 0, k ∈ {0, 1, . . .} and {B t , t ≥ 0} is a standard Brownian motion. Consider the one-dimensional controlled standard Brownian motion process {X t , t ≥ 0} defined by the stochastic differential equation dX t b0 XtkuXt dt dB t , 1.1 where u is the control variable, b0 > 0, k ∈ {0, 1, . In the case when k 0, Lefebvre and Whittle 1 were able to find the optimal control u∗ by making use of a theorem in Whittle 2, page 289 that enables us to express the value function. 1.4 u X t , 0≤t≤T x in terms of a mathematical expectation for the uncontrolled Brownian motion {B t , t ≥ 0} obtained by setting u ≡ 0 in 1.1. We cannot appeal to the theorem in Whittle 2 in that case, the author was able to express the function F x in terms of a mathematical expectation for an uncontrolled geometric Brownian motion.

Optimal Control

Maximal Value of λ

Conclusion