[Paper Review] Quadratic backward stochastic differential equations driven by $G$-Brownian motion: discrete solutions and approximation

Ying Hu, Yiqing Lin+1 more|arXiv (Cornell University)|Mar 11, 2016

Stochastic processes and financial applications33 references3 citations

TL;DR

This paper establishes the existence and uniqueness of solutions to quadratic backward stochastic differential equations (BSDEs) driven by $G$-Brownian motion, using discrete approximations via fully nonlinear PDEs and Banach space convergence. The key contribution is a rigorous framework for quadratic growth generators in the $G$-expectation setting, extending classical BSDE theory to Knightian uncertainty models.

ABSTRACT

In this paper, we consider backward stochastic differential equations driven by $G$-Brownian motion (GBSDEs) under quadratic assumptions on coefficients. We prove the existence and uniqueness of solution for such equations. On the one hand, a priori estimates are obtained by applying the Girsanov type theorem in the $G$-framework, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first construct solutions for discrete GBSDEs by solving corresponding fully nonlinear PDEs, and then approximate solutions for general quadratic GBSDEs in Banach spaces.

Motivation & Objective

• To establish the existence and uniqueness of solutions for backward stochastic differential equations driven by $G$-Brownian motion under quadratic growth conditions on the generator.
• To extend classical BSDE theory to the sublinear $G$-expectation framework, which models Knightian uncertainty in financial and stochastic models.
• To develop a discrete approximation scheme for $G$-BSDEs by solving fully nonlinear PDEs and proving convergence in Banach spaces.
• To overcome the lack of a priori estimates and uniqueness in the quadratic $G$-BSDE setting through Girsanov-type arguments and compactness methods.
• To generalize previous results on Lipschitz or bounded generators to the more complex case of quadratic growth in the control variable $Z$.

Proposed method

• Derive a priori estimates using a Girsanov-type theorem within the $G$-framework to establish uniqueness of solutions.
• Construct discrete solutions to $G$-BSDEs by solving associated fully nonlinear parabolic PDEs on a finite time grid.
• Use the Banach space framework to approximate the general quadratic $G$-BSDE by a sequence of discrete $G$-BSDEs.
• Apply the $G$-martingale representation theorem to decompose the solution into symmetric $G$-Itô integral and decreasing $G$-martingale parts.
• Employ modulus of continuity and concave majorants ($w^h$) to control the convergence of discrete approximations in sup-norm and $L^p$-norm.
• Leverage the Lebesgue dominated convergence theorem and properties of $G$-expectation to prove convergence of the approximating sequences in $\mathcal{S}^p_G$ and $\mathcal{H}^{p'}_G$.

Experimental results

Research questions

• RQ1 Can existence and uniqueness be established for quadratic $G$-BSDEs with generators having quadratic growth in $Z$?
• RQ2 How can discrete solutions to $G$-BSDEs be constructed using fully nonlinear PDEs?
• RQ3 What conditions ensure the convergence of discrete approximations to the true solution in the $G$-framework?
• RQ4 Can the $G$-martingale representation theorem be used to characterize the solution structure in the quadratic case?
• RQ5 What role does the $G$-expectation framework play in enabling solutions under model uncertainty?

Key findings

• The paper proves the existence and uniqueness of a solution $({Y}, {Z}, {K})$ to the quadratic $G$-BSDE in the space $\mathfrak{G}^2_G(0,T)$.
• A priori estimates are derived using a Girsanov-type transformation in the $G$-framework, which ensures uniqueness under quadratic growth.
• Discrete solutions are constructed by solving fully nonlinear PDEs corresponding to the $G$-BSDE on a time partition, ensuring well-posedness at each step.
• The sequence of discrete solutions converges in $\mathcal{S}^p_G(0,T) \times \mathcal{H}^{p'}_G(0,T) \times \mathcal{S}^{p''}_G(0,T)$ for arbitrarily large $p$, $p'$, $p''$, implying convergence in a strong sense.
• The limiting process $K$ is shown to be a decreasing $G$-martingale, consistent with the $G$-martingale representation theorem.
• The convergence of the approximating sequence is established via modulus of continuity estimates and the use of concave majorants $w^h$, ensuring $L^p$-convergence of $\bar{Y}^n$, $\bar{Z}^n$, and $\bar{K}^n$.