A class of Gaussian processes with fractional spectral measures

TL;DR

This paper investigates a family of stationary increment Gaussian processes defined by singular spectral measures, especially affine self-similar measures, extending stochastic calculus tools for these processes.

Contribution

It characterizes Gaussian processes with singular spectral measures, particularly affine self-similar measures, and develops Ito calculus for these processes using Kondratiev-white noise spaces.

Findings

Characterization of Gaussian processes with singular spectral measures

Extension of Ito stochastic calculus to these processes

Derivation of an associated Ito formula

Abstract

We study a family of stationary increment Gaussian processes, indexed by time. These processes are determined by certain measures sigma (generalized spectral measures), and our focus here is on the case when the measure sigma is a singular measure. We characterize the processes arising from when sigma is in one of the classes of affine self-similar measures. Our analysis makes use of Kondratiev-white noise spaces. With the use of a priori estimates and the Wick calculus, we extend and sharpen earlier computations of Ito stochastic integration developed for the special case of stationary increment processes having absolutely continuous measures. We further obtain an associated Ito formula.