Path-dependent Hamilton-Jacobi equations with super-quadratic growth in the gradient and the vanishing viscosity method

TL;DR

This paper advances the theory of path-dependent Hamilton-Jacobi equations with super-quadratic growth, establishing well-posedness, uniqueness, and a non-Markovian vanishing viscosity method for complex stochastic control problems.

Contribution

It introduces a non-Markovian vanishing viscosity approach and proves uniqueness of solutions for path-dependent viscous Hamilton-Jacobi equations with super-quadratic growth.

Findings

Proved uniqueness of maximal subsolutions.

Established well-posedness for path-dependent Hamilton-Jacobi-Bellman equations.

Demonstrated the applicability to stochastic control with state constraints.

Abstract

The non-exponential Schilder-type theorem in Backhoff-Veraguas, Lacker and Tangpi [Ann. Appl. Probab., 30 (2020), pp. 1321-1367] is expressed as a convergence result for path-dependent partial differential equations with appropriate notions of generalized solutions. This entails a non-Markovian counterpart to the vanishing viscosity method. We show uniqueness of maximal subsolutions for path-dependent viscous Hamilton-Jacobi equations related to convex super-quadratic backward stochastic differential equations. We establish well-posedness for the Hamilton-Jacobi-Bellman equation associated to a Bolza problem of the calculus of variations with path-dependent terminal cost. In particular, uniqueness among lower semi-continuous solutions holds and state constraints are admitted.