Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

Rainer Buckdahn; Christian Keller; Jin Ma; Jianfeng Zhang

doi:10.1186/s41546-020-00049-8

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2020, Volume 5: 7. Doi: 10.1186/s41546-020-00049-8

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Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

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

Christian Keller, christian.keller@ucf.edu

Received: March 10, 2020

Early access: November 2020

Published: September 2020

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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.

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Mathematics Subject Classification: 60H07;15;30;35R60;34F05.

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