[Paper Review] A class of globally solvable Markovian quadratic BSDE systems and applications

Hao Xing, Gordan Žitković|arXiv (Cornell University)|Mar 1, 2016

Stochastic processes and financial applications32 references19 citations

TL;DR

This paper establishes global existence and uniqueness for a broad class of Markovian systems of backward stochastic differential equations (BSDEs) with quadratic nonlinearities, using a novel Lyapunov function condition and local boundedness assumptions. The key contribution is a general solvability framework that applies to stochastic equilibria in incomplete markets, non-zero-sum stochastic games, and martingales on Riemannian manifolds without requiring smallness conditions on the terminal condition.

ABSTRACT

We establish existence and uniqueness for a wide class of Markovian systems of backward stochastic differential equations (BSDE) with quadratic nonlinearities. This class is characterized by an abstract structural assumption on the generator, an a-priori local-boundedness property, and a locally-Hölder-continuous terminal condition. We present easily verifiable sufficient conditions for these assumptions and treat several applications, including stochastic equilibria in incomplete financial markets, stochastic differential games, and martingales on Riemannian manifolds.

Motivation & Objective

• To resolve the long-standing open problem of global existence and uniqueness for multidimensional BSDEs with quadratic generators.
• To develop a general framework for solving Markovian quadratic BSDE systems without restrictive smallness conditions on the terminal condition.
• To establish sufficient conditions for solvability using a novel structural assumption—existence of a Lyapunov function—combined with local boundedness.
• To apply the framework to key problems in mathematical finance, stochastic games, and stochastic analysis on manifolds.
• To extend existing results by removing measure-change or diagonal-quadratic assumptions present in prior works.

Proposed method

• Introduces a new structural condition on the generator via the existence of a Lyapunov function, ensuring that h(Y) is a strict submartingale for any solution Y.
• Imposes an a-priori local-boundedness condition on the generator to control growth and ensure pathwise regularity.
• Uses a transformation of variables (e.g., Ŷ = Y¹ - Y²) to decouple the system and reduce it to a form amenable to analysis.
• Applies Theorem 2.14 to verify existence and uniqueness under conditions (BF) and (wAB), where (wAB) ensures boundedness via positively spanning vectors.
• Employs localization arguments and boundedness of the terminal condition in a local Hölder space to control the solution process.
• Verifies that the transformed generator satisfies the required conditions (BF) and (wAB) through explicit computation of quadratic forms and vector span conditions.

Experimental results

Research questions

• RQ1 Can global existence and uniqueness be established for Markovian systems of BSDEs with quadratic generators without smallness assumptions on the terminal condition?
• RQ2 What structural conditions on the generator ensure solvability in the multidimensional quadratic case?
• RQ3 How can Lyapunov functions be used to construct a priori estimates and ensure submartingale behavior in the solution process?
• RQ4 In what financial and geometric contexts do such systems naturally arise, and can they be solved globally?
• RQ5 Can the framework be extended to non-Markovian or superquadratic cases, or are there inherent limitations?

Key findings

• The system admits a unique bounded continuous solution whenever the generator satisfies the Lyapunov function condition and the a-priori local-boundedness condition.
• The solution exists for terminal conditions in a local Hölder space, without requiring smallness in L∞ norm, overcoming a major limitation in prior work.
• The existence of a Lyapunov function ensures that the solution process remains stochastically bounded, enabling the use of localization techniques.
• The framework applies to stochastic equilibria in incomplete financial markets, where the equilibrium condition reduces to a quadratic BSDE system.
• For non-zero-sum stochastic games, the method yields a Nash equilibrium with explicit value functions expressed as exponentials of solutions.
• The transformed system (5.13) satisfies conditions (BF) and (wAB), confirming existence and uniqueness via Theorem 2.14, even in the case θ > 1 with four spanning vectors.