Gaussian Processes with Volterra Kernels

Abstract

We study Volterra processes $X_t = \int _0^t K(t,s)dW_s$, where W is a standard Wiener process, and the kernel has the form $K(t,s) = a(s) \int _s^t b(u) c(u-s) du$. This form generalizes the Volterra kernel for fractional Brownian motion (fBm) with Hurst index $H>1/2$. We establish smoothness properties of X, including continuity and Hölder property. It happens that its Hölder smoothness is close to well-known Hölder smoothness of fBm but is a bit worse. We give a comparison with fBm for any smoothness theorem. Then we investigate the problem of inverse representation of W via X in the case where $c\in L^1[0,T]$ creates a Sonine pair, i.e. there exists $h\in L^1[0,T]$ such that $c * h \equiv 1$. It is a natural extension of the respective property of fBm that generates the same filtration with the underlying Wiener process. Since the inverse representations of the Gaussian processes under consideration are based on the properties of Sonine pairs, we provide several examples of Sonine pairs, both well-known and new.