Gâteaux type path-dependent PDEs and BSDEs with Gaussian forward processes

Abstract

We are interested in path-dependent semilinear PDEs, where the derivatives are of Gâteaux type in specific directions [Formula: see text] and [Formula: see text], being the kernel functions of a Volterra Gaussian process [Formula: see text]. Under some conditions on [Formula: see text] and the coefficients of the PDE, we prove existence and uniqueness of a decoupled mild solution, a notion introduced in a previous paper by the authors. We also show that the solution of the PDE can be represented through BSDEs where the forward (underlying) process is [Formula: see text].