A new integral equation for Brownian stopping problems with finite time horizon

TL;DR

This paper introduces a new Fredholm integral equation approach to characterize stopping boundaries in finite horizon Brownian stopping problems, providing analytical insights and potential numerical methods.

Contribution

It derives a novel integral equation for the stopping set and demonstrates its uniqueness and utility in analyzing boundary behavior near the terminal time.

Findings

Unique characterization of stopping boundaries via the integral equation

Analysis of boundary behavior close to terminal time

Potential for numerical implementation of the method

Abstract

For classical finite time horizon stopping problems driven by a Brownian motion \[V(t,x) = \sup_{t\leq\tau\leq0}E_{(t,x)}[g(\tau,W_{\tau})],\] we derive a new class of Fredholm type integral equations for the stopping set. For large problem classes of interest, we show by analytical arguments that the equation uniquely characterizes the stopping boundary of the problem. Regardless of the uniqueness, we use the representation to rigorously find the limit behavior of the stopping boundary close to the terminal time. Interestingly, it turns out that the leading-order coefficient is universal for wide classes of problems. We also discuss how the representation can be used for numerical purposes.