Constructing Time-Homogeneous Generalised Diffusions Consistent with Optimal Stopping Values

TL;DR

This paper addresses the inverse problem of reconstructing a diffusion process from a family of optimal stopping problem values, providing conditions for existence and uniqueness using generalized convexity techniques.

Contribution

It introduces a method to recover a diffusion process from optimal stopping values, characterizing when such a diffusion exists uniquely.

Findings

Existence and uniqueness conditions for the diffusion

Characterization of the diffusion via generalized convexity

Application to inverse optimal stopping problems

Abstract

Consider a set of discounted optimal stopping problems for a one-parameter family of objective functions and a fixed diffusion process, started at a fixed point. A standard problem in stochastic control/optimal stopping is to solve for the problem value in this setting. In this article we consider an inverse problem; given the set of problem values for a family of objective functions, we aim to recover the diffusion. Under a natural assumption on the family of objective functions we can characterise existence and uniqueness of a diffusion for which the optimal stopping problems have the specified values. The solution of the problem relies on techniques from generalised convexity theory