|Date: Wednesday, October 25, 2017
Location: 1866EH East Hall (3:00 PM to 4:00 PM)
Title: Viscosity solutions for fully nonlinear stochastic partial differential equations - a rough path view
Abstract: We study viscosity solutions for a fairly large class of fully nonlinear stochastic PDEs. Those equations can also be viewed as forward path-dependent PDEs and will be treated under a unified framework as rough PDEs. Our definition of viscosity solutions gathers the spirit of the previous notions that use test functions along stochastic characteristics, which are, in our setting, determined by a system of first-order rough differential equations. We show that such a definition is equivalent to the alternative one that uses semi-jets and prove basic properties such as consistency with classical solutions, stability, and a partial comparison principle for fully nonlinear rough PDEs under natural conditions. Furthermore, when the diffusion coefficient is semilinear, we establish a complete theory including (global) existence and a comparison principle by transforming the rough PDE into a standard PDE via the method of characteristics.
Joint work with Rainer Buckdahn, Jin Ma, and Jianfeng Zhang.
Speaker: Christian Keller