Abstract:

We provide a wellposdness theory of a class of a second order backward doubly stochastic differential equation (2BDSDEs). We prove existence and uniqueness of the solution under a Lipschitz type assumption on the generator, and we investigate the links between our 2BDSDEs and a class of parabolic fully non-linear Stochastic PDE's. Finally, we prove that the Markovian solution of 2BDSDEs provide aprobabilistic interpretation of the classical and stochastic viscosity solution of fully non-linear SPDEs. Similarly to Buckdahn and Ma (2001), we use theDoss-Sussnmann transformation to convert fully nonlinear SPDEs to fully nonlinear PDEs with random coefficients,then we use the solution of 2BDSDEs to provide the Feynman-Kac's formula. This talk is based on a joint work with AnisMatoussi (University of Le Mans and EcolePolytechnique) and Dylan Possamai (University of Paris dauphine).

[1] Matoussi,A., Possamai,D., Sabbagh,W.  Viscosity Solutions for Fully Nonlinear Stochastic PDEs via Second Order Backward doubly SDEs, forthcoming paper.