Itô's calculus under sublinear expectations via regularity of PDEs and rough paths

In this paper, we first study the martingale problem in a sublinear expectation space. The critical tool is the Evans-Krylov theorem on regularity properties for solutions of fully nonlinear PDEs. Based on the analysis for the martingale problem and inspired by the rough path theory, we then develop stochastic calculus with respect to a general stochastic process, and derive an Itô type formula and the integration-by-parts formula. Our framework is analytic in that it does not rely on the probabilistic concept of “independence” as in the G-expectation theory.