Path-dependent equations and viscosity solutions in infinite dimension

TL;DR

This paper establishes the first well-posedness results for viscosity solutions of path-dependent PDEs in infinite-dimensional Hilbert spaces, extending finite-dimensional theories to more complex, non-Markovian models.

Contribution

It introduces a novel well-posedness framework for viscosity solutions of PPDEs in infinite-dimensional spaces, addressing previously unresolved cases.

Findings

First well-posedness result for infinite-dimensional PPDEs

Applicability to equations beyond current finite-dimensional viscosity theory

Extension of path-dependent PDE analysis to Hilbert spaces

Abstract

Path-dependent PDEs (PPDEs) are natural objects to study when one deals with non Markovian models. Recently, after the introduction of the so-called pathwise (or functional or Dupire) calculus (see [15]), in the case of finite-dimensional underlying space various papers have been devoted to studying the well-posedness of such kind of equations, both from the point of view of regular solutions (see e.g. [15, 9]) and viscosity solutions (see e.g. [16]). In this paper, motivated by the study of models driven by path-dependent stochastic PDEs, we give a first well-posedness result for viscosity solutions of PPDEs when the underlying space is a separable Hilbert space. We also observe that, in contrast with the finite-dimensional case, our well-posedness result, even in the Markovian case, applies to equations which cannot be treated, up to now, with the known theory of viscosity solutions.