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.
Highlights
We study the fully nonlinear second-order stochastic partial differential equation (SPDE) du(t, x, ω) = f (t, x, ω, u, ∂x u, ∂x2x u) dt + g(t, x, ω, u, ∂x u) ◦ d Bt (1.1)with initial condition u(0, x, ω) = u0(x), where (t, x) ∈ [0, ∞) × R, B is a standard Brownian motion defined on a probability space (, F, P), f and g are FB-progressively measurable random fields, and ◦ denotes the Stratonovic integration.Our investigation will build on several aspects of the theories of pathwise solutions to SPDEs studied in the past two decades
We study fully nonlinear second-order stochastic PDEs
These include: the theory of stochastic viscosity solutions, initiated by Lions and Souganidis (1998a; 1998b; 2000a; 2000b) and studied by Buckdahn and Ma (2001a; 2001b; 2002); path-dependent PDEs (PPDEs) studied by Buckdahn et al (2015), based on the notion of path derivatives in the spirit of Dupire (2019); and the aspect of rough PDEs studied by Keller and Zhang (2016), in terms of the rough path theory (initiated by Lyons (1998)) and using the connection between Gubinelli’s derivatives for “controlled rough paths” (2004) and Dupire’s path derivatives
Summary
Our investigation will build on several aspects of the theories of pathwise solutions to SPDEs studied in the past two decades. These include: the theory of stochastic viscosity solutions, initiated by Lions and Souganidis (1998a; 1998b; 2000a; 2000b) and studied by Buckdahn and Ma (2001a; 2001b; 2002); path-dependent PDEs (PPDEs) studied by Buckdahn et al (2015), based on the notion of path derivatives in the spirit of Dupire (2019); and the aspect of rough PDEs studied by Keller and Zhang (2016), in terms of the rough path theory (initiated by Lyons (1998)) and using the connection between Gubinelli’s derivatives for “controlled rough paths” (2004) and Dupire’s path derivatives. The main purpose of this paper is to integrate all these notions into a unified framework, in which we shall investigate the most general well-posedness results for fully nonlinear SPDEs of the type (1.1)
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