Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs

Ibrahim Ekren,Jianfeng Zhang

doi:10.1186/s41546-016-0010-3

Abstract

In this paper we propose a new type of viscosity solutions for fully nonlinear path dependent PDEs. By restricting to certain pseudo Markovian structure, we remove the uniform non- degeneracy condition imposed in our earlier works [9, 10]. We establish the comparison principle under natural and mild conditions. Moreover, as applications 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.

Highlights

We study the following fully nonlinear parabolic path-dependent PDE with terminal condition u(T, ω) = ξ(ω): Lu(t, ω) := ∂t u(t, ω) + G(t, ω, u, ∂ωu, ∂ω2ωu) = 0, (t, ω) ∈ [0, T ) × . (1)
In a series of papers by Ekren et al (2014a) and Ekren et al (2016a; 2016b), we proposed a notion of a viscosity solution for such PPDEs and established its wellposedness: existence, comparison principle, and stability
In “Stochastic HJB equations” sections and “Path dependent Isaacs equation” we present two applications: the stochastic HJB equations induced from the optimization problem with random coefficients and the path-dependent Bellman-Isaacs equations induced from the zero sum stochastic differential games

Summary

Introduction

Note that the path frozen PDE (3) is a local PDE, with the domain Qn induced from the stopping times Hn. in the degenerate case, the Hn used in (Ekren et al 2016a, 2016b) has very bad regularity, and the PDE (3) in Qn typically does not have a continuous viscosity solution. In order to construct smooth test functions, we let Gε be a smooth mollifier of G and require the following mollified path frozen PDE (with smooth boundary condition) has a classical solution: As we see in the following example, in the degenerate case such a PDE with a smooth boundary condition may not have a continuous viscosity solution.

Objectives

Results

Conclusion