Doubly Reflected Backward SDEs Driven by $G$-Brownian Motion with Quadratic Generator

Abstract: In this paper, we study the doubly reflected backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs for short) when the generator has quadratic growth in the $z$-component. Based on the theory of $G$-BMO martingale and $G$-Girsanov theorem, we establish the existence and uniqueness result when the upper obstacle is almost a generalized $G$-Itô's process. Moreover, the solution can be approximated monotonically by the solutions to a family of penalized reflected $G$-BSDEs with a lower obstacle, which plays an important role to establish the relation between doubly reflected $G$-BSDEs and fully nonlinear partial differential equations with double obstacles.

• The paper establishes the existence and uniqueness of solutions for doubly reflected G-BSDEs with quadratic generators under the G-expectation framework.
• Penalization techniques and uniform a priori estimates are employed to overcome challenges arising from double reflection and quadratic growth.
• The study connects stochastic solutions to viscosity solutions of nonlinear PDEs, offering insights for robust financial modeling and control.

Doubly Reflected Backward SDEs Driven by $G$-Brownian Motion with Quadratic Generator

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Background and Motivation

Backward stochastic differential equations (BSDEs) and their reflected and doubly reflected extensions with specific growth conditions on their generators constitute a significant component in the intersection of stochastic analysis, control theory, and the probabilistic representation of partial differential equations (PDEs). The $G$-expectation framework, introduced by Peng, enables the robust modeling of volatility uncertainty by replacing classical Brownian motion with $G$-Brownian motion and substituting linear expectation with sublinear expectation, encapsulating model ambiguity.

Doubly reflected BSDEs (DRBSDEs) provide a stochastic formulation for Dynkin games and related optimal stopping problems, particularly in nonlinear expectation environments. While much attention has been paid to Lipschitz generators, practical applications in finance and robust control motivate the study of quadratic generators. The combination of quadratic growth in $z$, double reflection, and sublinear $G$-expectation substantially increases analytic and probabilistic complexity, primarily due to the lack of a linear theory and the subtleties of $G$-martingales.

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Main Contributions

This work establishes the existence and uniqueness of solutions to doubly reflected $G$-BSDEs with a generator of quadratic growth in $z$. The authors extend the theory beyond the previously considered singly reflected or Lipschitz generator cases, addressing several technical obstacles unique to the $G$-framework:

1. Well-posedness under Quadratic Growth: Existence and uniqueness are proven when the upper obstacle is an almost generalized $G$-Itô process and the generator exhibits quadratic growth in $G$0. The solution inhabits the space $G$1.
2. Penalization and Monotone Approximation: Solutions are constructed as limits of penalized reflected $G$2-BSDEs, with a carefully designed sequence ensuring monotonic convergence—crucial for both analytic purposes and for establishing links to double-obstacle PDEs.
3. A Priori and Uniform Estimates: The development of uniform a priori estimates for the relevant processes underlines the results. The analysis resolves key difficulties where, unlike the lower obstacle case, estimates for $G$3 and the $G$4-martingale components can no longer be controlled solely by those of $G$5 due to the competitive effect of double penalization.
4. Probabilistic Representation for Nonlinear PDEs: The solution to the doubly reflected $G$6-BSDE is connected to the viscosity solution of fully nonlinear parabolic PDEs with two obstacles via a nonlinear Feynman-Kac formula within the Markovian setting. The construction of a monotone convergence scheme is essential for the validity of this representation.

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Core Theory and Techniques

Problem Formulation

Let $G$7 denote the terminal condition, generator functions, and lower/upper obstacles, respectively, all adapted to the $G$8-expectation framework. The doubly reflected $G$9-BSDE is defined as the following:

$G$0

subject to the constraints $G$1 $G$2, and an approximate Skorohod condition (AMC), which generalizes the minimally active push at the boundaries in the $G$3-setting and for quadratic growth.

Penalization Approach

Solutions are approximated using penalized reflected equations, where the penalization terms are designed to strongly discourage the solution from crossing the obstacles. The convergence analysis leverages stability results for quadratic $G$4-BSDEs, delicate comparison theorems, and $G$5-BMO techniques to ensure uniform bounds and tightness in necessary topologies.

Specifically, for integers $G$6 and $G$7, the sequence $G$8 is constructed to solve

$G$9

where $z$0 and $z$1 represent aggregated penalization for breach of the lower and upper obstacles, respectively. The limit as $z$2 recovers the solution to the original doubly reflected $z$3-BSDE.

BMO and Girsanov Techniques

Uniqueness and key estimates rely on meticulous application of the $z$4-BMO martingale framework and the $z$5-Girsanov transformation, extending the classical tools to the non-linear expectation setting. These are crucial for controlling exponential martingales and ensuring integrability and norm bounds on the solution components under iterated $z$6-expectations.

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Strong Numerical and Analytical Outcomes

• Uniform $z$7 Estimate: For the penalized solutions, tight uniform bounds in the $z$8-supremum norm are established, independent of penalization parameters, which is nontrivial due to the quadratic generator.
• Quantitative Convergence Rates: For $z$9, explicit rates of decay in $G$0 are provided, differentiating the analysis from earlier work and enabling precise control as penalization strengthens.
• Viscosity Solution Representation: The solution to the penalized $G$1-BSDEs is shown to converge monotonically to the solution of a fully nonlinear double-obstacle PDE via a Feynman-Kac correspondence. The technical machinery ensures the transition from probabilistic to analytic domains is valid under minimal regularity assumptions.

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Implications and Future Directions

Practically, the results broaden the stochastic tools available for robust pricing and hedging in financial markets with volatility uncertainty—especially in American and game options where double reflection and pathwise constraints appear. Theoretically, the work sets a precedent for analyzing broader classes of semi-linear or fully nonlinear obstacle problems under sublinear expectations and uncertain volatility.

The technical approach, notably the deployment of $G$2-BMO controls and penalized monotone schemes, opens avenues for:

• Extending to higher-dimensional $G$3-Brownian frameworks.
• Handling more general growth conditions or allowing for random/irregular obstacles.
• Developing numerical schemes for DRBSDEs under $G$4-expectation, leveraging the established monotonic convergence and uniform estimates.
• Studying optimal switching under model uncertainty, given the close relationship to double obstacle problems.

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Conclusion

This paper provides a rigorous and comprehensive theory for doubly reflected $G$5-BSDEs with quadratic generators, bridging the gap between robust stochastic processes and the analytic theory of fully nonlinear PDEs with double obstacles. By solving key technical challenges in the non-linear expectation and quadratic growth settings, the authors both clarify the foundational landscape and set the stage for future theoretical, computational, and applied advances in areas of uncertainty quantification and stochastic control.

Reference: "Doubly Reflected Backward SDEs Driven by $G$6-Brownian Motion with Quadratic Generator" (2604.23656)