Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

Abstract

Abstract We study fully nonlinear second-order (forward) stochastic PDEs. They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework. For the most general fully nonlinear case, we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. If the diffusion coefficient is semilinear (i.e, linear in the gradient of the solution and nonlinear in the solution; the drift can still be fully nonlinear), we establish a complete theory, including global existence and a comparison principle.