Volterra differential equations with singular kernels

TL;DR

This paper investigates Volterra stochastic differential equations with singular kernels, motivated by applications to fractional Brownian motion, focusing on existence, uniqueness, and properties of solutions without assuming kernel boundedness.

Contribution

It introduces a framework for analyzing Volterra SDEs with singular kernels, extending previous results to cases lacking boundedness assumptions.

Findings

Established existence and uniqueness of solutions

Characterized properties of solutions with singular kernels

Extended analysis to non-bounded kernel cases

Abstract

Motivated by the potential applications to the fractional Brownianmotion, we study Volterra stochasticdifferential of the form~:\begin{equation}X\_t = x+ \int\_0^tK(t,s)b(s,X\_s)ds + \int\_0^tK(t,s) \sigma(s,X\_s)\,dB\_s ,\tag{E} \label{eq:sdefbm}\end{equation}where $(B\_s, \, s\in [0,1])$ is a one-dimensional standard Brownianmotion and $(K(t,s), \, t,s \in [0,1])$ is a deterministic kernelwhose properties will be precised below but for which we don't assumeany boundedness property.