On Wald Optimal Stopping Problem for Geometric Brownian Motions

Abstract

This note concerns a problem of optimally stopping a nondegenerate, two-dimensional, geometric Brownian motion Q t = (x t ,y t ), with the goal of maximizing where the supremum is taken over the class of all stopping times 𝒳 Q , with finite expectation, H:ℝ+2 → ℝ is a measurable function satisfying a certain growth condition, and c > 0 is a positive constant. It is proved that, under certain conditions, the maximal value Φ(.,.) is a logarithmic function, and the optimal stopping time τ* < ∞ admits the form where ψ( · ) ∈ C 2(0,∞), positive non-decreasing solution of a certain second-order nonlinear ordinary differential equation. The present result extends and supplements a class of Wald-type optimal stopping problems in Graversen and Peskir's paper (1997), which treats the case of a one-dimensional Brownian motion.