Dr. Jianfeng Zhang (USC)

Title: Viscosity Solutions of Fully Nonlinear Parabolic Path-Dependent PDEs

Abstract: In this talk we introduce a notion of viscosity solutions for Path Dependent PDEs (PPDEs for short). Such new type of PDEs include, among others: Backward SDEs (semilinear PPDE), G-martingales and Second Order BSDEs (Path dependent HJB equations), Path dependent Bellman-Issacs equation (corresponding to zero-sum stochastic differential game), Backward Stochastic PDEs, and (Forward) Stochastic PDEs. Compared to the standard approach in the literature on those topics, a viscosity theory enables us to: (i) mimic the PDE language to characterize the solutions to those SDEs; (ii) provide a unified treatment to all those subjects and extend to even more general framework; (iii) establish the wellposedness result (existence, uniqueness, comparison, stability) under weaker conditions.
A PPDE uses continuous paths as its variables, and thus is an infinite dimensional problem and lacks local compactness, a crucial property used in the literature of standard viscosity theory to prove the comparison principle. To overcome this main difficulty, we decompose the comparison principle into a partial comparison principle and a Perron’s type of arguments. Such an approach seems new and we do not use the Ishii’s lemma directl.