Stochastic Volterra equation

Summary

A stochastic Volterra equation is a Volterra integral equation driven by noise. It has the form $X(t) = X_0(t) + \int_0^t a(t, s, X(s)) \, ds + \int_0^t b(t, s, X(s)) \, dW(s)$, where the last integral is an Itô integral against a Wiener process. When the coefficients $a$ and $b$ do not depend on $t$ and $X_0$ is constant, it reduces to a stochastic differential equation. The dependence on $t$ lets the noise and the drift have memory. Such memory is used in models of rough volatility in finance and of viscoelastic materials under random forcing.