Optimal stopping for L\'evy processes and affine functions

TL;DR

This paper investigates optimal stopping problems for Lévy processes, providing a new method to determine optimal thresholds without complex calculations, and establishing the convexity and structure of the value function.

Contribution

It introduces a novel approach to find optimal thresholds in Lévy process stopping problems, simplifying the analysis by avoiding complex integro-differential calculations.

Findings

The smallest optimal stopping time is a hitting time.

The value function is convex.

A new method to determine optimal thresholds efficiently.

Abstract

This paper studies an optimal stopping problem for L\'evy processes. We give a justification of the form of the Snell envelope using standard results of optimal stopping. We also justify the convexity of the value function, and without a priori restriction to a particular class of stopping times, we deduce that the smallest optimal stopping time is necessarily a hitting time. We propose a method which allows to obtain the optimal threshold. Moreover this method allows to avoid long calculations of the integro-differential operatorused in the usual proofs.