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  ^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.

Keywords:

• Geometric Brownian motions
• Smooth-fit principle
• Wald's identity

Subject Classification:

• 60G40
• 62L15

ACKNOWLEDGMENTS

We acknowledge with many thanks to the Editor-in-Chief, Professor Nitis Mukhopadhyay, and the anonymous referee for their constructive and useful comments, which led to an improved version of our earlier submission. We also thank the referee for calling our attention to Lai's paper (2004), which contains some recent results on Wald identities. Partial results of this note were obtained when the author was at the Mathematics Institute, University of Oslo (Norway), under the postdoc grant PRO12/1003.

Notes

Recommended by Nitis Mukhopadhyay