Risk-sensitive optimal stopping with unbounded terminal cost function

TL;DR

This paper studies risk-sensitive optimal stopping problems over an infinite horizon with unbounded costs, revealing multiple solutions to the Bellman equation and providing methods to approximate them through finite horizon problems.

Contribution

It introduces a probabilistic interpretation for multiple solutions of the Bellman equation in unbounded cases and proposes approximation techniques using finite horizon problems.

Findings

Multiple solutions to the Bellman equation can exist in unbounded cases.

Probabilistic interpretation of minimal and maximal solutions is provided.

Finite horizon approximations effectively estimate the solutions.

Abstract

In this paper we consider an infinite time horizon risk-sensitive optimal stopping problem for a Feller--Markov process with an unbounded terminal cost function. We show that in the unbounded case an associated Bellman equation may have multiple solutions and we give a probabilistic interpretation for the minimal and the maximal one. Also, we show how to approximate them using finite time horizon problems. The analysis, covering both discrete and continuous time case, is supported with illustrative examples.