Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

Probability, Uncertainty and Quantitative Risk

Christian Keller¹  Jianfeng Zhang²  Jin Ma²  Rainer Buckdahn³ 

[1] Department of Mathematics, University of Central Florida, Orlando, Florida, United States;Department of Mathematics, University of Southern California, Los Angeles, California, United States;Univ Brest, UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique, Brest, France, and Shandong University, Jinan, China;

关键词: Stochastic PDEs;    path-dependent PDEs;    rough PDEs;    rough paths;    viscosity solutions;    comparison principle;    functional Itô formula;    characteristics;    rough Taylor expansion;    60H07;    15;    30;    35R60;    34F05;   

DOI  :  10.1186/s41546-020-00049-8

来源: Springer

【 摘 要 】

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.

【 授权许可】

CC BY