Weak solutions to distribution-dependent stochastic Volterra equations

TL;DR

This paper establishes the existence of weak solutions for a class of distribution-dependent stochastic Volterra equations, including those with singular kernels, under certain growth and regularity conditions.

Contribution

It introduces a novel approach linking local martingale problems to weak solutions for these equations, broadening the scope of solvable models.

Findings

Proves existence of weak solutions under linear growth and continuity conditions.

Establishes a connection between local martingale problems and weak solutions.

Derives continuity and integrability properties of solutions.

Abstract

We prove the existence of weak solutions for distribution-dependent stochastic Volterra equations under linear growth and continuity conditions on the coefficients and mild regularity assumptions on the kernels, including singular kernels. To this end, we formulate an associated local martingale problem and establish its connection with weak solutions. Moreover, we derive continuity and integrability properties of the solutions.