On the optimality of threshold type strategies in single and recursive optimal stopping under L\'evy models

TL;DR

This paper proves the optimality of threshold strategies in single and recursive optimal stopping problems under Lévy models with continuous additive functional discounting, providing new conditions and simpler proofs for threshold properties.

Contribution

It introduces an average problem approach to establish threshold optimality in Lévy models, extending to recursive problems with simplified proofs and broader applicability.

Findings

Optimal threshold strategies are proven for Lévy models with CAF discounting.

Conditions on reward functions and discounting are specified for spectrally negative models.

Simpler proofs are provided for qualitative properties of optimal thresholds.

Abstract

In the spirit of [Surya07'], we develop an average problem approach to prove the optimality of threshold type strategies for optimal stopping of L\'evy models with a continuous additive functional (CAF) discounting. Under spectrally negative models, we specialize this in terms of conditions on the reward function and random discounting, where we present two examples of local time and occupation time discounting. We then apply this approach to recursive optimal stopping problems, and present simpler and neater proofs for a number of important results on qualitative properties of the optimal thresholds, which are only known under a few special cases.