A Parametric Methodology for $G$-Expectations and Stochastic Systems under Model Uncertainty

This paper develops a systematic parametric method for analyzing stochastic systems under volatility uncertainty within the $G$-expectation framework. Leveraging the dual representation of the $G$-expectation as a supremum over a family of probability measures, we operationalize the correspondence between the $G$-expectation space and a parameterized family of classical stochastic processes. Our approach decomposes complex nonlinear analysis into two distinct phases: a ``linear implementation phase,'' which utilizes classical stochastic analysis under a fixed parameter, and a ``consistent estimation phase'' across the parameter space. We rigorously prove that this parametric mapping preserves fundamental stochastic structures, including the It\^o integral and quadratic variation, thus enabling a direct transplantation of classical It\^o calculus techniques into the $G$-framework. The advantages of this methodology are illustrated through its application to $G$-stochastic differential equations and $G$-backward stochastic differential equations. Furthermore, we demonstrate that this method effectively mitigates the inherent accumulation of estimation errors found in traditional stepwise sublinear procedures, yielding significantly more accurate estimates for fundamental quantities in robust stochastic analysis.