Optimal stopping under nonlinear expectation

Ibrahim Ekren, Nizar Touzi, Jianfeng Zhang

Research output: Contribution to journalArticlepeer-review


Let X:[0,T]×ΩR be a bounded càdlàg process with positive jumps defined on the canonical space of continuous paths Ω. We consider the problem of optimal stopping the process X under a nonlinear expectation operator E 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 E-supermartingale, and an E-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. [6,7]).

Original languageEnglish (US)
Pages (from-to)3277-3311
Number of pages35
JournalStochastic Processes and their Applications
Issue number10
StatePublished - Oct 2014


  • Keywords
  • Nonlinear expectation
  • Optimal stopping
  • Snell envelope

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


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