Probability, Uncertainty and Quantitative Risk | |
Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions | |
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
【 预 览 】
Files | Size | Format | View |
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RO202104281467054ZK.pdf | 1100KB | download |