Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

Abstract

We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We develop first a local theory of classical solutions and define then viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative one using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. When the diffusion coefficient is semi-linear (but the drift can be fully nonlinear), we establish a complete theory, including global existence and comparison principle. Our methodology relies heavily on the method of characteristics.

Crandall-Lions Viscosity Solutions for Path-Dependent PDEs: The Case of Heat Equation

“…We also mention [4], where such a minimax approach was extended to first-order path-dependent Hamilton-Jacobi-Bellman equations in infinite dimension.Concerning the second-order case, a first attempt to extend the Crandall-Lions framework to the path-dependent case was carried out in [74], even though a technical condition on the semi-jets was imposed, namely condition ( 16) in [74], which narrows down the applicability of such a result. In the literature, this was perceived as an almost insurmountable obstacle, so that the Crandall-Lions definition was not further investigated, while other notions of generalized solution were devised, see [32,75,89,18,58,3,8]. We mention in particular the framework designed in [32] and further investigated in [33,34,78,79,80,15], where the notion of sub/supersolution adopted differs from the Crandall-Lions definition as the tangency condition is not pointwise but in the sense of expectation with respect to an appropriate class of probability measures.…”