Gaussian processes with Volterra kernels

Yuliya Mishura, Georgiy Shevchenko, Sergiy Shklyar

Published 2020-01-10Version 1

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 Holder property. It happens that its Holder smoothness is close to well-known Holder 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 $c * h = 1$. It is a natural extension of the respective property of fBm that generates the same filtration with the underlying Wiener process. Since the inverse representation 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. Key words: Gaussian process, Volterra process, Sonine pair, continuity, Holder property, inverse representation.