Pathwise Patching: A Geometric Construction of Quasi-Sure Solutions to G-SDEs

This paper explores a geometric approach to constructing quasi-sure solutions for $G$-stochastic differential equations (G-SDEs) under model uncertainty. We propose a pathwise patching methodology that systematically combines measure-specific solutions into a unified, universally measurable process. The construction relies on pathwise uniqueness and the convex structure of the $G$-expectation framework to ensure compatibility across different probability measures. We further investigate the possibility of developing a robust Malliavin calculus within this framework. By reformulating the Malliavin derivative through variational equations that inherit the quasi-sure structure, we attempt to overcome the challenges posed by traditional $L^2$-closure arguments in multi-measure settings. While our approach offers enhanced geometric intuition and a potentially more transparent foundation for stochastic analysis under model uncertainty, we recognize its limitations and the need for further investigation into its full scope and applicability. The results may have implications for robust financial mathematics and stochastic control, though we present them as a preliminary step toward more comprehensive theories.