Optimal stopping under nonlinear expectation

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:124

Article

Ekren, Ibrahim¹  Touzi, Nizar²  Zhang, Jianfeng¹ 

[1] Univ So Calif, Dept Math, Los Angeles, CA 90089 USA

[2] Ecole Polytech Paris, CMAP, Paris, France

关键词: Nonlinear expectation;    Optimal stopping;    Snell envelope;   

DOI  :  10.1016/j.spa.2014.04.006

来源: Elsevier

【 摘 要 】

Let X : [0, T] x Omega -> R be a bounded cadlag process with positive jumps defined on the canonical space of continuous paths Omega. We consider the problem of optimal stopping the process X under a nonlinear expectation operator epsilon defined as the supremum of expectations over a weakly compact but nondominated family of probability 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 epsilon-supermartingale, and an epsilon-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. (2014), in the semilinear case, and extended to the fully nonlinear case in the accompanying papers (Ekren et al. [6,71). (C) 2014 Elsevier B.V. All rights reserved.

【 授权许可】

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