A stochastic calculus for Rosenblatt processes

Abstract

A stochastic calculus is given for processes described by stochastic integrals with respect to fractional Brownian motions and Rosenblatt processes somewhat analogous to the stochastic calculus for Itô processes. These processes for this stochastic calculus arise naturally from a stochastic chain rule for functionals of Rosenblatt processes; and some Itô-type expressions are given here. Furthermore, there is some analysis of these results for their applications to problems using Rosenblatt noise.

On Advances in Noise Modeling in Stochastic Systems, Control and Adaptive Control

“…Some of the stochastic calculus for Rosenblatt processes is given in Refs. [11,15]. A change of variables (Itô-Doeblin) formula is given in Čoupek et al [11] that is used here to determine explicit Radon-Nikodym derivatives analogous to an approach for Wiener measure.…”

“…[11,15]. A change of variables (Itô-Doeblin) formula is given in Čoupek et al [11] that is used here to determine explicit Radon-Nikodym derivatives analogous to an approach for Wiener measure. The subsequent application of the change of variables formula for Rosenblatt processes uses the following two differential operators,…”

“…where Wt ðÞis a standard Wiener process. The change of variables results for stochastic equations driven by Rosenblatt processes is described now following [11]. Consider a scalar process yt ðÞ , t ≥ 0 ðÞ for the description of the change of variables result.…”

“…The solution of an ergodic control problem for a scalar linear system with a Rosenblatt noise is also given [6]. Stochastic calculus methods were developed for fractional Brownian motion [7-10] and for Rosenblatt processes [11]. Stochastic calculus provides powerful tools: stochastic integrals, Itô's differential formula, and martingales.…”

Optimal controls for fractional stochastic differential systems driven by Rosenblatt process with impulses

“…The Rosenblatt process is a non‐Gaussian process with many interesting properties such as stationarity of the increments, long‐range dependence, and self‐similarity. For more recent works on the Rosenblatt process, see References 17‐21 and the references therein. Shen et al 22 analyzed the controllability and stability of fractional stochastic functional systems driven by Rosenblatt process.…”