Optimal Stopping with Random Maturity under Nonlinear Expectations

TL;DR

This paper studies an optimal stopping problem with random maturity under nonlinear expectations, allowing for jumps and control over drift and volatility, extending classical models to more complex stochastic settings.

Contribution

It introduces a framework for optimal stopping with random maturity under a set of singular probability measures, including jump processes and controlled stochastic dynamics.

Findings

Formulation of the stopping problem under nonlinear expectations and singular measures

Analysis of the problem with jumps and controlled drift and volatility

Extension of classical stopping theory to more general stochastic models

Abstract

We analyze an optimal stopping problem with random maturity under a nonlinear expectation with respect to a weakly compact set of mutually singular probabilities $\mathcal{P}$. The maturity is specified as the hitting time to level $0$ of some continuous index process at which the payoff process is even allowed to have a positive jump. When $\mathcal{P}$ is a collection of semimartingale measures, the optimal stopping problem can be viewed as a {\it discretionary} stopping problem for a player who can influence both drift and volatility of the dynamic of underlying stochastic flow.