Path independence for the additive functionals of stochastic Volterra equations with singular kernels and H\"older continuous coefficients

TL;DR

This paper establishes the well-posedness and path-independence of additive functionals for stochastic Volterra equations with singular kernels and H"older continuous coefficients, linking them to stochastic PDEs and fractional Brownian motion.

Contribution

It introduces a novel approach to characterize path-independence for non-semimartingale stochastic Volterra equations via stochastic PDEs.

Findings

Proves well-posedness of stochastic Volterra equations with singular kernels.

Characterizes path-independence for additive functionals of these equations.

Extends results to equations driven by fractional Brownian motion kernels.

Abstract

In this paper, we are concerned with stochastic Volterra equations with singular kernels and H\"older continuous coefficients. We first establish the well-posedness of these equations by utilising the Yamada-Watanabe approach. Then, we aim to characterise the path-independence for additive functionals of these equations. The main challenge here is that the solutions of stochastic Volterra equations are not semimartingales nor Markov processes, thus the existing techniques for obtaining the path-independence of usual, semimartingale type stochastic differential equations are no longer applicable. To overcome this difficulty, we link the concerned stochastic Volterra equations to mild formulation of certain parabolic type stochastic partial differential equations, and further apply our previous results on the path-independence for stochastic evolution equations to get the desired result. Finally, as an important application, we consider a class of stochastic Volterra equations whose kernels are related with fractional Brownian motions and derive the path-independence of additive functionals for them.