Parameter dependent optimal thresholds, indifference levels and inverse optimal stopping problems

TL;DR

This paper develops a general theory for parameter-dependent infinite-horizon optimal stopping problems with threshold strategies, introducing an indifference map that generalizes the Gittins index and enables inverse problem solutions.

Contribution

It introduces a supermodularity-based framework for analyzing parameter dependence and indexability in infinite-horizon stopping problems, extending to inverse optimal stopping problems.

Findings

Supermodularity guarantees indexability of the problem family.

The indifference map generalizes the classical Gittins index.

Framework enables solving inverse optimal stopping problems.

Abstract

Consider the classic infinite-horizon problem of stopping a one-dimensional diffusion to optimise between running and terminal rewards and suppose we are given a parametrised family of such problems. We provide a general theory of parameter dependence in infinite-horizon stopping problems for which threshold strategies are optimal. The crux of the approach is a supermodularity condition which guarantees that the family of problems is indexable by a set valued map which we call the indifference map. This map is a natural generalisation of the allocation (Gittins) index, a classical quantity in the theory of dynamic allocation. Importantly, the notion of indexability leads to a framework for inverse optimal stopping problems.