Optimal stopping under adverse nonlinear expectation and related games

TL;DR

This paper establishes the existence of optimal stopping times and saddle points in zero-sum games involving nonlinear expectations, with applications to American option subhedging under volatility uncertainty.

Contribution

It introduces a nonlinear Snell envelope framework and proves the existence of optimal strategies in a class of games with sublinear expectations, including G-expectation.

Findings

The game has a well-defined value.

Optimal stopping times are characterized by hitting times of the nonlinear Snell envelope.

Results apply to American option subhedging under volatility uncertainty.

Abstract

We study the existence of optimal actions in a zero-sum game $\inf_{\tau}\sup_PE^P[X_{\tau}]$ between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem $\inf_{\tau}\mathcal{E}(X_{\tau})$ for a class of sublinear expectations $\mathcal{E}(\cdot)$ such as the $G$-expectation. We show that the game has a value. Moreover, exploiting the theory of sublinear expectations, we define a nonlinear Snell envelope $Y$ and prove that the first hitting time $\inf\{t:Y_t=X_t\}$ is an optimal stopping time. The existence of a saddle point is shown under a compactness condition. Finally, the results are applied to the subhedging of American options under volatility uncertainty.