Stochastic Volterra equations with random functional coefficients in Banach spaces

TL;DR

This paper develops a method to solve stochastic Volterra equations with random, possibly singular, coefficients in Banach spaces, ensuring solutions are well-defined and measurable.

Contribution

It introduces a novel approach to establish unique Banach-valued solutions for stochastic Volterra equations with complex, random coefficients, including distribution-dependent cases.

Findings

Solutions are unique and Banach-valued.

Solutions are strongly measurable under certain conditions.

The method handles singular kernels and distribution-dependent coefficients.

Abstract

We derive unique Banach-valued solutions to stochastic Volterra equations with random coefficients that may depend on pure chance and involve singular kernels. In particular, for controlled and distribution-dependent coefficients these solutions become strong, as a measurability analysis of the Wasserstein metric confirms. The presented novel approach is based on the proof that a stochastic Volterra integral admits a progressively measurable modification in a weak sense and on sharp moment estimates for non-negative product measurable processes.