On the regularity of solutions of some linear parabolic path-dependent PDEs

TL;DR

This paper investigates the regularity and existence of solutions for a class of linear parabolic path-dependent PDEs, extending previous work by considering more general coefficient dependencies and establishing new existence and uniqueness results.

Contribution

It generalizes existing results on PPDEs by allowing coefficients to depend on integrals of paths, and proves the existence of smooth solutions under broader conditions.

Findings

Existence of smooth solutions under ellipticity and Hölder conditions

Extension of regularity results to more general path dependencies

Existence and uniqueness of weak solutions for related path-dependent SDEs

Abstract

We study a class of linear parabolic path-dependent PDEs (PPDEs) defined on the space of c\`adl\`ag paths $x \in D([0,T])$, in which the coefficient functions at time $t$ depend on $x(t)$ and $\int_{0}^{t}x(s)dA_{s}$, for some (deterministic) continuous function $A$ with bounded variations. Under uniform ellipticity and H\"older regularity conditions on the coefficients, together with some technical conditions on $A$, we obtain the existence of a smooth solution to the PPDE by appealing to the notion of Dupire's derivatives. It provides a generalization to the existing literature studying the case where $A_t = t$, and complements our recent work, Bouchard and Tan (2021), on the regularity of approximate viscosity solutions for parabolic PPDEs. As a by-product, we also obtain existence and uniqueness of weak solutions for a class of path-dependent SDEs.