学术报告（张建丰 10.08）

Abstract:  Path dependent PDEs considers continuous paths as its variable. It is a convenient tool for stochastic optimization/games in non-Markovian setting, and has natural applications on non-Markovian financial models with drift and/or volatility uncertainty. For example, a martingale can be viewed as a solution to a path dependent heat equation, and we are particularly interested in path dependent HJB equations and Isaacs equations. In path dependent case, even a heat equation typically does not have a classical solution, where smoothness is defined through Dupire's functional Ito calculus, so a viscosity theory is desirable. However, here the state space (of continuous paths) is not locally compact, a crucial property used in the standard viscosity theory in PDE literature. To get around of this difficulty, our main innovation is to replace the pointwise maximum/minimum in the definition of PDE viscosity solution with an optimal stopping problem. In this talk, we will motivate our definition of viscosity solution by focusing on heat equations, and then establish the wellposedness of fully nonlinear PPDEs: existence, comparison and uniqueness, and stability. The talk is based on a series of works joint with Ibrahim Ekren, Christian Keller, Zhenjie Ren, and Nizar Touzi.