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Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs

Research supported in part by NSF grant DMS 1413717.
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  • 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.
    Mathematics Subject Classification: 35D40;35K10;60H10;60H30.


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