# Paracontrolled distribution approach to stochastic Volterra equations

## TL;DR

This paper introduces a paracontrolled distribution framework to establish existence and uniqueness for rough Volterra equations with singular kernels, broadening the scope of stochastic differential equations that can be analyzed.

## Contribution

It develops a novel paracontrolled distribution approach for rough Volterra equations and introduces convolutional rough paths applicable to various stochastic processes.

## Key findings

- Existence and uniqueness results for rough Volterra equations with singular kernels.
- Construction of convolutional rough paths for processes like fractional Brownian motion.
- Application to stochastic differential equations with delay and driven by Gaussian or Lévy processes.

## Abstract

Based on the notion of paracontrolled distributions, we provide existence and uniqueness results for rough Volterra equations of convolution type with potentially singular kernels and driven by the newly introduced class of convolutional rough paths. The existence of such rough paths above a wide class of stochastic processes including the fractional Brownian motion is shown. As applications we consider various types of rough and stochastic (partial) differential equations such as rough differential equations with delay, stochastic Volterra equations driven by Gaussian processes and moving average equations driven by L\'evy processes.