A weak solution theory for stochastic Volterra equations of convolution type

TL;DR

This paper develops a weak solution framework for stochastic Volterra equations with jumps, accommodating non-Lipschitz coefficients and singular kernels, using weak convergence and martingale problem techniques.

Contribution

It introduces a novel approach combining a priori Sobolev-Slobodeckij estimates and a new martingale problem to establish existence, stability, and uniqueness of solutions.

Findings

Established weak existence and stability results for stochastic Volterra equations

Proved path regularity and uniqueness under additional conditions

Applied results to nonlinear Hawkes processes and Markovian approximations

Abstract

We obtain general weak existence and stability results for stochastic convolution equations with jumps under mild regularity assumptions, allowing for non-Lipschitz coefficients and singular kernels. Our approach relies on weak convergence in $L^p$ spaces. The main tools are new a priori estimates on Sobolev-Slobodeckij norms of the solution, as well as a novel martingale problem that is equivalent to the original equation. This leads to generic approximation and stability theorems in the spirit of classical martingale problem theory. We also prove uniqueness and path regularity of solutions under additional hypotheses. To illustrate the applicability of our results, we consider scaling limits of nonlinear Hawkes processes and approximations of stochastic Volterra processes by Markovian semimartingales.