Optimal Stopping under Nonlinear Expectation

Abstract

Let $X$ be a bounded càdlàg process with positive jumps defined on the canonical space of continuous paths. We consider the problem of optimal stopping the process $X$ under a nonlinear expectation operator $\cE$ defined as the supremum of expectations over a weakly compact family of nondominated measures. We introduce the corresponding nonlinear Snell envelope. Our main objective is to extend the Snell envelope characterization to the present context. Namely, we prove that the nonlinear Snell envelope is an $\cE-$supermartingale, and an $\cE-$martingale up to its first hitting time of the obstacle $X$. This result is obtained under an additional uniform continuity property of $X$. We also extend the result in the context of a random horizon optimal stopping problem. This result is crucial for the newly developed theory of viscosity solutions of path-dependent PDEs as introduced in Ekren et al., in the semilinear case, and extended to the fully nonlinear case in the accompanying papers (Ekren, Touzi, and Zhang, parts I and II).

36 pages

References in corpus (3)

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• A Complete Representation Theorem for $G$-martingales
• Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation
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• Path-Dependent Optimal Stochastic Control and Viscosity Solution of Associated Bellman Equations
• On the convergence of monotone schemes for path-dependent PDE
• Some Norm Estimates for Semimartingales
• Monotone Schemes for Fully Nonlinear Parabolic Path Dependent PDEs