Reflected Stochastic Differential Equations Driven by G-Brownian Motion with Nonlinear Constraints

Abstract: In this paper, we study the reflected stochastic differential equations driven by G-Brownian motion (reflected G-SDEs) with two nonlinear constraints. With the help of the Skorokhod problem with nonlinear constraints, we first study the doubly reflected G-Brownian motion, which is constructed pathwise and lies in the same G-expectation space as the G-Brownian motion. For the reflected G-SDE, the uniqueness is derived from some a priori estimate and the existence is obtained by a Picard iteration method. The comparison theorem of the solution and the individual constraining processes are provided.

• The paper establishes the existence and uniqueness of reflected G-SDEs under relaxed, nonlinear constraint conditions.
• It employs a hybrid analytic and pathwise method to solve the Skorokhod problem under model uncertainty with minimal reflection intervention.
• The framework extends G-Itô calculus, offering robust solutions for finance and risk management applications under volatility ambiguity.

Reflected $G$-Stochastic Differential Equations with Nonlinear Constraints: Existence, Uniqueness, and Pathwise Construction

Introduction and Motivation

This work addresses the solvability of reflected stochastic differential equations (SDEs) driven by $G$-Brownian motion (hereafter, reflected $G$-SDEs) subject to two nonlinear constraints. The context is the $G$-expectation framework introduced by Peng, which enables robust stochastic analysis under model uncertainty, encapsulating volatility ambiguity via a sublinear expectation. Classical reflected SDEs and associated Skorokhod problems have been widely studied, including those with time-dependent and nonlinear boundaries on classical probability spaces. However, most research in this direction assumes bi-Lipschitz or linear constraint functions, and none adequately capture the challenges posed by model uncertainty and nonlinear constraint regimes. The present paper systematically develops both the underlying Skorokhod problem for these constraints and the well-posedness theory for the corresponding forward SDEs, all under the $G$-expectation.

Skorokhod Problem with Nonlinear Constraints Beyond Bi-Lipschitz Regimes

The Skorokhod problem is revisited in settings where the two constraint functions, $g$ and $h$, may not be bi-Lipschitz. Instead, the authors assume strict monotonicity and invertibility conditions with a strictly positive gap between boundaries but relax Lipschitz continuity. This generalization is significant: previous results, e.g., [Li 2023], require bi-Lipschitz assumptions that exclude common nonlinearities.

The paper establishes that under the new set of weaker conditions, the Skorokhod problem associated with a given input path $s$ and constraints $g,h$ possesses a unique solution $(x,k)$. Furthermore, the constraining process $G$0 allows explicit pathwise characterization via running suprema and infima of transformed input and boundary maps. Continuity results with respect to both input perturbations and constraint function variations are also derived, quantifying the stability of reflected paths and constraining processes under all admissible inputs.

$G$1-Expectation Framework and Technical Foundations

The $G$2-expectation construction involves a canonical process (the $G$3-Brownian motion $G$4), with the underlying expectation being a supremum over a non-dominated set of measures parametrized by volatility bounds. The authors carefully recall the relevant function spaces, quasi-sure analysis, and the characterization of essential spaces such as $G$5, $G$6, and $G$7. The $G$8-Itô and Itô–Tanaka formulae extend to this setting, permitting stochastic calculus with integrators of bounded variation, which is crucial for the construction of solutions to reflected SDEs and handling the constraining processes.

A notable technical result is that pathwise compositions involving running suprema and infima of $G$9-valued processes remain in $G$0. This enables pathwise realization of constraining processes even when boundaries are input-dependent and stochastic within the $G$1-framework.

Existence and Uniqueness for Reflected $G$2-SDEs Under Nonlinear Constraints

The core technical contribution is the demonstration of existence and uniqueness for reflected $G$3-SDEs with general Lipschitz coefficients and strictly monotone, invertible (but not necessarily Lipschitz) nonlinear constraints. The equation under consideration has the form:

$G$4

Existence is established using a Picard iteration, iterating between a non-reflected $G$5-SDE and the Skorokhod problem to enforce constraints at each step. Uniqueness follows from a-priori $G$6 estimates in the $G$7-framework and stability/control-theoretic arguments, even with path-dependent or nonlinear boundaries.

A strong feature of the analysis is the pathwise realization of the constraining process even under model uncertainty: for almost all sample paths in the quasi-sure sense, the reflecting process maintains minimal intervention, thus non-proliferation of the reflection measure, which is captured via the generalized Skorokhod condition.

Comparison Theorems and Monotonicity Results

The paper establishes a comprehensive comparison principle for both the solutions and the constraining processes of doubly reflected $G$8-SDEs. For solutions corresponding to different sets of coefficients and constraint functions, if the drift and volatility coefficients, as well as the inverse constraint functions, satisfy suitable monotonicity relations, then the ordering of the solutions is preserved quasi surely. The results extend and refine classical comparison theorems by accounting for the joint effects of nonlinear, possibly random, constraints and nonstandard drift/diffusion terms under $G$9-expectation.

Moreover, the study provides monotonicity relations for the constraining processes themselves: greater lower constraints or smaller upper constraints require increased intervention by the respective constraining process, quantified via explicit bounds in terms of the data and initial values. This insight is nontrivial and significant when modeling constrained systems under ambiguity.

Technical Advancements: Pathwise Construction Under $G$0-Uncertainty

Key advancements include:

• A pathwise construction of solutions (including constraining processes) that is robust to both model uncertainty and nonlinearity in constraints;
• Demonstration that minimality of the Skorokhod reflection persists in the $G$1-expectation setting;
• The extension of $G$2-Itô calculus to stochastic calculus with respect to general bounded variation (and nondecreasing) constraining processes;
• Derivation of sharp a-priori estimates for both solution trajectories and constraint interventions.

No numerical simulations are presented, but the theoretical development is precise and extends the scope of $G$3-SDE theory.

Practical and Theoretical Implications

Practically, these results expand the modeling capacity for robust control and finance problems (e.g., pricing under volatility uncertainty with regulatory constraints, constrained portfolio evolution under Knightian uncertainty, systemic risk evaluation under ambiguous evolution laws) where state variables are required to remain within nonlinear, possibly random regimes. The minimal reflection principle ensures no extraneous adjustment is applied beyond what is strictly necessary to maintain valid states. The framework readily generalizes to multidimensional and coupled systems.

Theoretically, the relaxation of constraint regularity assumptions is nontrivial and removes barriers to applying the G-framework in more realistic models, since most real-world constraints are not globally Lipschitz or linear. The pathwise approach opens avenues for further stochastic analysis under sublinear expectation, including backward SDEs, fully nonlinear PDEs, and stochastic control under ambiguity.

Conclusion

This work rigorously extends reflected SDE theory to cover $G$4-Brownian motion under general nonlinear constraints without the bi-Lipschitz restriction. Existence, uniqueness, stability, and comparison theorems are established via a hybrid of analytic and pathwise arguments in the $G$5-expectation setting. The framework is robust to model uncertainty and provides a foundation for further study of constrained systems under volatility ambiguity, reflected backward SDEs, and related fields. The results are anticipated to have impact across robust finance, risk management, and the mathematics of uncertainty quantification.

Reference: "Reflected Stochastic Differential Equations Driven by G-Brownian Motion with Nonlinear Constraints" (2604.22130).