Markovian Lifts of Stochastic Volterra Equations in Sobolev Spaces: Solution theory, an Ito Formula and Invariant Measures

TL;DR

This paper develops a solution framework for Markovian lifts of stochastic Volterra equations with monotone kernels in Sobolev spaces, introduces an Ito formula, and studies invariant measures of these processes.

Contribution

It provides a comprehensive solution theory for non-local stochastic evolution equations derived from Volterra equations, including conditions for invariant measures and an Ito formula.

Findings

Established existence of invariant measures for the lifted processes

Developed an Ito-type formula for the considered stochastic Volterra equations

Provided a new solution framework in weighted Sobolev spaces

Abstract

We investigate Markovian lifts of stochastic Volterra equations (SVEs) with completely monotone kernels and general coefficients within a class of weighted Sobolev spaces. Our primary focus is developing a comprehensive solution theory for a class of non-local stochastic evolution equations (SEEs) encompassing these Markovian lifts. This enables us to provide conditions for the existence of invariant measures for the lifted processes and the corresponding SVE. Another key contribution is an Ito-type formula for the stochastic Volterra equations under consideration.