The solution of discretionary stopping problems with applications to the optimal timing of investment decisions

TL;DR

This paper introduces a comprehensive methodology for explicitly solving infinite horizon optimal stopping problems involving one-dimensional Itô diffusions, broad payoff functions, and state-dependent discounting, with applications in finance and economics.

Contribution

It develops a unified framework combining dynamic programming, variational inequalities, and probabilistic methods to solve a wide class of optimal stopping problems explicitly.

Findings

Explicit solutions for a broad class of stopping problems

Framework applicable to finance and economic decision-making

Bridges PDE and probabilistic approaches for optimal stopping

Abstract

We present a methodology for obtaining explicit solutions to infinite time horizon optimal stopping problems involving general, one-dimensional, It\^o diffusions, payoff functions that need not be smooth and state-dependent discounting. This is done within a framework based on dynamic programming techniques employing variational inequalities and links to the probabilistic approaches employing $r$-excessive functions and martingale theory. The aim of this paper is to facilitate the the solution of a wide variety of problems, particularly in finance or economics.