Optimal Stopping under Nonlinear Expectation

By Ibrahim Ekren, Nizar Touzi, Jianfeng Zhang. 2012

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Let $X$ be a bounded c\`adl\`ag 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).

Subfields: Economics and Econometrics, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/b11915d00c68ca46ad46b9b8a6a9dd68226c3ca5

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Citations in the corpus

Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II (2012, background): “This paper is the continuation of our accompanying papers [7, 8].”
Path-Dependent Optimal Stochastic Control and Viscosity Solution of Associated Bellman Equations (2012, methodology): “[45] H. M. Soner, N. Touzi, and J. Zhang.”
Optimal stopping under adverse nonlinear expectation and related games (2012, methodology): “Our construction uses a compactness argument and a result of [9] on the approximation of hitting times by random times that are continuous in ω to find a measure P which is optimal up to the time τ ∗.”
A complete representation theorem for G-martingales (2012, background): “The theory has received very strong attention in the literature in recent years, we refer to the survey paper [20] and the references therein as well as some more recent developments: [4,5,10,12,13,15,16,26], to mention a few.”