[Paper Review] A Generalized Mixed Zero-sum Stochastic Differential Game and Double Barrier Reflected BSDEs with Quadratic Growth Coefficient

Saïd Hamadène, Eduard Rotenstein+1 more|arXiv (Cornell University)|Jul 9, 2008

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes the existence of a saddle point in a generalized mixed zero-sum stochastic differential game where payoffs are modeled by doubly reflected backward stochastic differential equations (BSDEs) with quadratic growth in the Z-component. It proves that the upper and lower value functions are deterministic and uniquely characterized as viscosity solutions to two Isaacs equations with obstacles, extending risk-sensitive control and American game option frameworks.

ABSTRACT

This article is dedicated to the study of mixed zero-sum two-player stochastic differential games in the situation when the player's cost functionals are modeled by doubly controlled reflected backward stochastic equations with two barriers whose coefficients have quadratic growth in Z. This is a generalization of the risk-sensitive payoffs. We show that the lower and the upper value function associated with this stochastic differential game with reflection are deterministic and they are also the unique viscosity solutions for two Isaacs equations with obstacles.

Motivation & Objective

• To generalize risk-sensitive control and American game options by modeling payoffs through doubly reflected BSDEs with quadratic growth in the Z-component.
• To analyze mixed zero-sum stochastic differential games where players can stop early and face penalties.
• To establish the existence and uniqueness of upper and lower value functions as deterministic viscosity solutions to Isaacs equations with two obstacles.
• To unify the frameworks of Dynkin games and risk-sensitive control under a common BSDE-based structure with reflection and quadratic coefficients.

Proposed method

• The authors model the game's payoff using a doubly reflected BSDE with two barriers and coefficients exhibiting quadratic growth in the Z-component.
• They employ Kobylanski's existence and uniqueness result for BSDEs with quadratic growth to ensure well-posedness of the underlying BSDE.
• The value functions are derived from the solution of the BSDE via an exponential transformation, linking the BSDE solution to the game's payoff structure.
• The connection between the game's value and the BSDE solution is formalized using Itô's formula and martingale representation, showing that the payoff equals the exponential of the initial BSDE value.
• Viscosity solution theory is applied to characterize the upper and lower value functions as solutions to two Isaacs equations with obstacles.
• The proof leverages the characterization of BSDE solutions with two reflecting barriers to derive dynamic programming principles for the value functions.

Experimental results

Research questions

• RQ1 Can the value functions of a mixed zero-sum stochastic differential game with stopping times and reflection be characterized as viscosity solutions to a system of HJB-type equations with obstacles?
• RQ2 How does the inclusion of quadratic growth in the Z-component of the generator affect the existence and uniqueness of solutions to the underlying BSDE?
• RQ3 Under what conditions do the upper and lower value functions coincide, and what is their deterministic nature in this generalized game setting?
• RQ4 To what extent does the exponential payoff structure with risk-sensitive preferences generalize standard risk-neutral Dynkin games and American game options?
• RQ5 Is the solution of the doubly reflected BSDE with quadratic growth sufficient to represent the value of a mixed stochastic differential game with stopping times?

Key findings

• The upper and lower value functions of the mixed zero-sum stochastic differential game are deterministic, which is a key structural result.
• These value functions are uniquely characterized as viscosity solutions to two Isaacs equations with two obstacles, extending classical HJB theory to the reflected and quadratic-growth setting.
• The payoff of the game is represented as the exponential of the initial value of a doubly reflected BSDE with quadratic growth in Z, linking stochastic control to BSDE theory.
• The solution of the BSDE with two reflecting barriers and quadratic generator ensures the existence and uniqueness of the value functions under boundedness and regularity assumptions on the coefficients.
• The dynamic programming principle for the value functions is established through the characterization of the BSDE solution using essential infima and suprema over stopping times.
• The paper identifies a gap in the literature: the equality of upper and lower values remains an open problem, despite the existence of both value functions as viscosity solutions.