On the existence of weak solutions to stochastic Volterra equations

TL;DR

This paper proves the existence of weak solutions for a broad class of stochastic Volterra equations with time-inhomogeneous coefficients and general kernels, using a novel local martingale problem approach.

Contribution

It introduces a new method based on a local martingale problem to establish weak solutions for stochastic Volterra equations with general kernels.

Findings

Weak solutions exist for equations with time-inhomogeneous coefficients.

Applicable to equations with convolutional or bounded kernels in the diffusion term.

The approach broadens the class of stochastic Volterra equations with known solution existence.

Abstract

The existence of weak solutions is established for stochastic Volterra equations with time-inhomogeneous coefficients allowing for general kernels in the drift and convolutional or bounded kernels in the diffusion term. The presented approach is based on a newly formulated local martingale problem associated to stochastic Volterra equations.