[Paper Review] Optimal Stopping under G-expectation

Hanwu Li|arXiv (Cornell University)|Dec 20, 2018

Stochastic processes and financial applications9 references4 citations

TL;DR

This paper establishes a theory of optimal stopping under G-expectation, introducing G-stopping times to handle Knightian uncertainty, particularly volatility ambiguity. It proves the value function is the minimal G-supermartingale dominating the payoff process and links it to reflected G-BSDEs, extending results from discrete to continuous and infinite time horizons with existence of optimal stopping times.

ABSTRACT

We develop a theory of optimal stopping problems under G-expectation framework. We first define a new kind of random times, called G-stopping times, which is suitable for this problem. For the discrete time case with finite horizon, the value function is defined backwardly and we show that it is the smallest G-supermartingale dominating the payoff process and the optimal stopping time exists. Then we extend this result both to the infinite horizon and to the continuous time case. We also establish the relation between the value function and solution of reflected BSDE driven by G-Brownian motion.

Motivation & Objective

• To develop a theory of optimal stopping under G-expectation to model Knightian uncertainty, especially volatility ambiguity.
• To define a new class of random times, G-stopping times, that preserve time consistency and conditional G-expectation properties.
• To extend the classical optimal stopping framework—where value functions are minimal supermartingales—to the nonlinear G-framework.
• To establish a connection between the value function and solutions of reflected backward stochastic differential equations (BSDEs) driven by G-Brownian motion.
• To prove existence of optimal stopping times and characterize the value function in discrete, infinite, and continuous time settings.

Proposed method

• Introduce G-stopping times as a generalization of stopping times suitable for G-expectation, ensuring well-defined conditional G-expectations.
• Define the value function backwardly in discrete finite-horizon settings, showing it is the minimal G-supermartingale dominating the payoff process.
• Extend the value function to infinite horizon via limits of finite-horizon value functions, preserving the minimal G-supermartingale property.
• Establish a connection between the value function and the solution of a reflected G-BSDE with a lower obstacle, where the solution Y is a G-supermartingale dominating the obstacle process X.
• Use the existence and uniqueness theory of G-BSDEs and reflected G-BSDEs (from Hu, Ji, Peng, Song) to justify the solution structure.
• Leverage the Girsanov transformation and comparison theorem for G-BSDEs to analyze time consistency and optimality.

Experimental results

Research questions

• RQ1 How can optimal stopping be formulated under G-expectation when classical stopping times fail due to lack of quasi-sure measurability and conditional expectation issues?
• RQ2 What is the appropriate generalization of stopping times in the G-framework that preserves time consistency and enables backward induction?
• RQ3 Is the value function in the G-framework still the minimal G-supermartingale dominating the payoff process, as in the classical case?
• RQ4 Can the value function be characterized as the solution of a reflected G-BSDE, and what is the role of the reflected process in optimality?
• RQ5 How do the results extend from finite to infinite and continuous time horizons under G-expectation?

Key findings

• The value function in the discrete finite-horizon case is the smallest G-supermartingale dominating the payoff process and is obtained via backward induction.
• The value function in the infinite-horizon case is defined as the limit of finite-horizon value functions and remains the minimal G-supermartingale dominating the payoff process.
• The optimal stopping time exists and is a G-stopping time, with the value process being a G-martingale up to the stopping time.
• The value function coincides with the first component Y of the solution to a reflected G-BSDE with obstacle process X, where Y is a G-supermartingale dominating X.
• The solution to the reflected G-BSDE exists and is unique under standard conditions on the generator, obstacle, and terminal condition, with Y ∈ S_G^α(0,T) for 2 ≤ α < β.
• The minimal G-supermartingale property of the value function is preserved under the G-framework, even under volatility uncertainty.