Parameterized Controls and Isomorphic Representations of G-Expectations

Abstract: This paper establishes a fundamental isomorphism between the abstract framework of $G$-expectation and a concrete parameterized family of classical stochastic processes, providing a unified framework for nonlinear stochastic calculus under model uncertainty. We demonstrate that $G$-Brownian motion and related stochastic differential equations (SDEs) can be intrinsically represented as controlled processes ${B_t^\theta}_{\theta \in \mathcal{A}^\Theta}$, where $\theta$ denotes an admissible control process taking values in a compact convex parameter set $\Theta$ encoding volatility uncertainty. The core contribution is the construction of a complete algebraic and analytic isomorphism $\Phi$ that preserves all operations including addition, multiplication, function composition, and limit transitions. This isomorphism provides a rigorous mathematical foundation for understanding $G$-martingales, with a distinction between symmetric martingales (fair in all volatility scenarios) and asymmetric martingales (favorable but fair only in the worst-case scenario). We develop a comprehensive theory for solving $G$-SDEs through their parameterized representations, establishing existence and uniqueness results under standard Lipschitz and growth conditions. The framework reveals how bounded control sets preserve these classical conditions while capturing volatility uncertainty. The parameterized control perspective offers both mathematical rigor and computational practicality, enabling the transfer of classical stochastic calculus results to the $G$-expectation framework while maintaining the essential sublinear structure of model uncertainty.