The authors propose results of investigation of properties of Gaussian processes admitting the integral representation via the Wiener process. They deal with the processes represented in the form \(X_t=\int_0^t K(t,s) dW_s\), where \(W_s\) is a standard Wiener process, the Volterra kernel \(K(t,s)\) is of the form \(K(t,s)=a(s)\int^t_s b(u)c(u-s)du\), and \(a,b,c:[0,T]\to\mathbb R\) are measurable functions. This form is a natural generalization of the fractional Brownian motion (fBm), which admits the integral representation via the Wiener process, and the Volterra kernel of its representation consists of power functions.
The authors describe the properties of the functions \(a,b,c\), which provide a certain smoothness of the process \(X_t\), including continuity and Hölder property. Then, the authors describe properties of the functions \(a,b,c\), which guarantee that the Wiener process and the corresponding Gaussian process generate the same filtration. It is turned out that the functions form the so-called Sonine pair (see [N. Ja. Sonin and N. I. Akhiezer (ed.), Untersuchungen über Zylinderfunktionen und spezielle Polynome. Moskau: Staatsverlag für technisch-theoretische Literatur (1954; Zbl 0056.06502)]), property that the components of the kernel generating fBm have. Finally, the authors provide some examples of Sonine pairs to illustrate the described property.
For the entire collection see [Zbl 1515.60023].