Viscosity solutions of path dependent PDEs

 时间：7月14日--17日 13:3-16:00

摘要：Path dependent PDEs (PPDEs, for short) 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 BSDE can be viewed as a solution to a semilinear PPDE, 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, in this setting the state space (of continuous paths) is not locally compact, which is 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 short course, we will motivate our definition of viscosity solution by focusing on heat equations, and then establish the wellposedness for fully nonlinear PPDEs: existence, comparison and uniqueness, and stability.