Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs

In this paper,we propose a new type of viscosity solutions for fully nonlinear path-dependent PDEs.By restricting the solution to a pseudo-Markovian structure defined below,we remove the uniform non-degeneracy condition needed in our earlier works(Ekren,I,Touzi,N,Zhang,J,Ann Probab,44:1212–1253,2016a;Ekren,I,Touzi,N,Zhang,J,Ann Probab,44:2507–2553,2016b)to establish the uniqueness result.We establish the comparison principle under natural and mild conditions.Moreover,we apply our results to two important classes of PPDEs:the stochastic HJB equations and the path-dependent Isaacs equations,induced from the stochastic optimization with random coefficients and the path-dependent zero-sum game problem,respectively.