Weak well-posedness of stochastic Volterra equations with completely monotone kernels and non-degenerate noise

TL;DR

This paper proves weak existence and uniqueness for a class of stochastic Volterra equations with singular kernels and multiplicative noise, demonstrating a regularization-by-noise effect through reformulation as a stochastic evolution equation.

Contribution

It introduces a novel approach to establish weak well-posedness for SVEs with completely monotone kernels and non-degenerate noise by reformulating them as stochastic evolution equations.

Findings

Weak existence and uniqueness in law for SVEs with singular kernels.

Regularization-by-noise effect demonstrated for SVEs with multiplicative noise.

Reformulation of SVEs into stochastic evolution equations enables the analysis.

Abstract

We establish weak existence and uniqueness in law for stochastic Volterra equations (SVEs for short) with completely monotone kernels and non-degenerate noise under mild regularity assumptions. In particular, our results reveal the regularization-by-noise effect for SVEs with singular kernels, allowing for multiplicative noise with H\"{o}lder diffusion coefficients. In order to prove our results, we reformulate the SVE into an equivalent stochastic evolution equation (SEE for short) defined on a Gelfand triplet of Hilbert spaces. We prove weak well-posedness of the SEE using stochastic control arguments, and then translate it into the original SVE.