Strict comparison theorems under sublinear expectations

In this paper, we consider strict comparison theorems in the framework of G-expectation, which is a type of sublinear expectation associated with fully nonlinear parabolic partial differential equations. In particular, we first apply Krylov–Safonov estimates to establish the strict comparison theorem for functions from the Lipschitz class (Lip(Omega )). Then we prove generalized strict comparison theorems on the enlarged space (L_G^1(Omega )), which is the Banach completion of (Lip(Omega )) under the G-expectation.