Volterra differential equations with singular kernels

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) σ(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.

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• Large and moderate deviations for stochastic Volterra systems