Path-dependent SDEs in Hilbert spaces

TL;DR

This paper investigates path-dependent stochastic differential equations in Hilbert spaces, establishing existence, uniqueness, and differentiability of solutions with respect to initial data and system parameters using contraction methods.

Contribution

It introduces a novel framework for analyzing path-dependent SDEs in Hilbert spaces, including differentiability and stability results for mild solutions.

Findings

Proved existence and uniqueness of mild solutions.

Established Gâteaux differentiability of solutions.

Demonstrated continuity of derivatives with respect to data.

Abstract

We study path-dependent SDEs in Hilbert spaces. By using methods based on contractions in Banach spaces, we prove existence and uniqueness of mild solutions, continuity of mild solutions with respect to perturbations of all the data of the system, G\^ateaux differentiability of generic order n of mild solutions with respect to the starting point, continuity of the G\^ateaux derivatives with respect to all the data. The analysis is performed for generic spaces of paths that do not necessarily coincide with the space of continuous functions.