Arr\^et optimal pour les processus de Markov forts et les fonctions affines

TL;DR

This paper investigates optimal stopping problems for strong Markov processes with affine functions, establishing the form of the Snell envelope, the convexity of the value function, and characterizing the minimal optimal stopping time as a hitting time.

Contribution

It provides a rigorous justification of the Snell envelope form and the convexity of the value function without restricting to specific stopping times, identifying the smallest optimal stopping time as a hitting time.

Findings

Snell envelope form justified using standard optimal stopping results

Convexity of the value function established

Smallest optimal stopping time characterized as a hitting time

Abstract

In this Note we study optimal stopping problems for strong Markov processes and affine functions. We give a justification of the Snell envelope form 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.