Abstract
The path dependent PDEs (PPDE, for short) is a powerful and convenient tool for non-Markov problems. In particular, a BSDE can be viewed as a semilinear PPDE and thus the nonlinear Feynman-Kac formula is extended to non-Markov case. In this chapter we are most interested in fully nonlinear parabolic PPDEs, typically path dependent Hamilton-Jacobi-Bellman equations and path dependent Bellman-Isaacs equations, due to their important applications in stochastic control and stochastic differential games. However, in path dependent case, even a heat equation may not have a classical solution. We thus turn to viscosity solutions. We shall introduce a new notion of viscosity solution and establish its well-posedness, including the comparison principle.
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