Optimal stopping in a general framework

TL;DR

This paper develops a general framework for optimal stopping problems involving families of non-negative random variables, establishing existence, characterization, and properties of optimal stopping times under weak assumptions.

Contribution

It introduces a broad, flexible approach to optimal stopping, relaxing regularity and integrability conditions, and provides new existence and characterization results.

Findings

Existence of an optimal stopping time under weak assumptions.

Characterization of minimal and maximal optimal stopping times.

Analysis of local properties of the value function family.

Abstract

We study the optimal stopping time problem $v(S)={\rm ess}\sup_{\theta \geq S} E[\phi(\theta)|\mathcal {F}_S]$, for any stopping time $S$, where the reward is given by a family $(\phi(\theta),\theta\in\mathcal{T}_0)$ \emph{of non negative random variables} indexed by stopping times. We solve the problem under weak assumptions in terms of integrability and regularity of the reward family. More precisely, we only suppose $v(0) < + \infty$ and $ (\phi(\theta),\theta\in \mathcal{T}_0)$ upper semicontinuous along stopping times in expectation. We show the existence of an optimal stopping time and obtain a characterization of the minimal and the maximal optimal stopping times. We also provide some local properties of the value function family. All the results are written in terms of families of random variables and are proven by only using classical results of the Probability Theory.