[Paper Review] Optimal Stopping under Nonlinear Expectation

Ibrahim Ekren, Nizar Touzi+1 more|arXiv (Cornell University)|Sep 28, 2012

Stochastic processes and financial applications11 references18 citations

TL;DR

This paper establishes the nonlinear Snell envelope characterization for optimal stopping under nonlinear expectation, proving it is an ${\cal E}$-supermartingale and an ${\cal E}$-martingale up to the first hitting time of the obstacle. The key contribution is a novel limiting argument using quasi-continuous approximations to overcome the failure of the dominated convergence theorem in nonlinear expectation frameworks, enabling extension of classical optimal stopping theory to nondominated, singular measures.

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).

Motivation & Objective

• To extend the classical Snell envelope characterization to optimal stopping under nonlinear expectation with a nondominated family of singular measures.
• To establish that the nonlinear Snell envelope is an ${\cal E}$-supermartingale and an ${\cal E}$-martingale up to the first hitting time of the obstacle $X$.
• To overcome the failure of the dominated convergence theorem in nonlinear expectation by constructing quasi-continuous approximations for sequences of stopping times.
• To support the viscosity solution theory for path-dependent PDEs by providing a probabilistic foundation for fully nonlinear cases.

Proposed method

• Define the nonlinear expectation ${\cal E}[\cdot] := \sup_{\mathbb{P} \in \mathcal{P}} \mathbb{E}^\mathbb{P}[\cdot]$ over a weakly compact family $\mathcal{P}$ of nondominated singular measures.
• Introduce the nonlinear Snell envelope $Y$ as the minimal ${\cal E}$-supermartingale dominating $X_{\tau \wedge \textsc{h}}$.
• Prove the ${\cal E}$-supermartingale property via a dynamic programming principle adapted to nonlinear expectations.
• Establish the ${\cal E}$-martingale property on $[0, \tau^*]$ using a limiting argument on decreasing sequences $Y_{\tau_n}$, relying on the monotone convergence theorem for quasi-continuous functions from Denis, Hu, and Peng [3].
• Construct quasi-continuous approximations of $Y_{\tau_n}$ to apply the monotone convergence theorem despite the lack of dominated convergence in nonlinear frameworks.
• Use weak compactness of $\mathcal{P}$ to ensure convergence of expectations and control error terms in iterative stopping time constructions.

Experimental results

Research questions

• RQ1 Can the classical Snell envelope characterization be extended to optimal stopping under nonlinear expectation with nondominated measures?
• RQ2 How can the ${\cal E}$-martingale property be established on $[0, \tau^*]$ when the dominated convergence theorem fails?
• RQ3 What role do quasi-continuous approximations play in enabling convergence arguments under nonlinear expectation?
• RQ4 How does the uniform continuity of the process $X$ facilitate the construction of regular value processes in the absence of smoothness?
• RQ5 What is the connection between the nonlinear Snell envelope and the viscosity solution theory for path-dependent PDEs?

Key findings

• The nonlinear Snell envelope $Y$ is an ${\cal E}$-supermartingale, generalizing the classical result to nonlinear expectation frameworks.
• The nonlinear Snell envelope $Y$ is an ${\cal E}$-martingale up to the first hitting time $\tau^*$ of the obstacle $X$, ensuring optimality of $\tau^*$.
• The failure of the dominated convergence theorem is overcome by constructing quasi-continuous approximations of $Y_{\tau_n}$, enabling use of the monotone convergence theorem.
• The proof relies crucially on the weak compactness of the family $\mathcal{P}$ to control error terms and ensure convergence in expectation.
• The result provides a probabilistic foundation for viscosity solutions of fully nonlinear path-dependent PDEs, extending prior work in the semilinear case.
• The iterative construction of stopping times $\tau^n$ with controlled error terms leads to $\mathbb{E}^{\mathbb{P}^0}[\widehat{Y}_{\tau^0}] \leq \mathbb{E}^{\mathbb{P}^m}[\widehat{Y}_{\tau^m} \mathbf{1}_{D_m} + \widehat{Y}_{\widehat{\tau}^*} \mathbf{1}_{D_m^c}] + C\bar{\rho}_0(3\delta) + 4\varepsilon$, which implies the desired inequality in the limit.