Spectral functions related to some fractional stochastic differential equations

TL;DR

This paper investigates fractional higher-order stochastic differential equations driven by Gaussian white noise, deriving explicit covariance and spectral functions for the solutions, thus advancing understanding of their spectral properties.

Contribution

The paper introduces explicit solutions and spectral functions for a class of fractional higher-order stochastic differential equations, expanding analytical tools in stochastic process theory.

Findings

Explicit covariance functions derived

Spectral functions explicitly obtained

Enhanced understanding of fractional stochastic processes

Abstract

In this paper we consider fractional higher-order stochastic differential equations of the form \begin{align*} \left( \mu + c_\alpha \frac{d^\alpha}{d(-t)^\alpha} \right)^\beta X(t) = \mathcal{E}(t) , \quad t\geq 0,\; \mu>0,\; \beta>0,\; \alpha \in (0,1) \cup \mathbb{N} \end{align*} where $\mathcal{E}(t)$ is a Gaussian white noise. We derive stochastic processes satisfying the above equations of which we obtain explicitly the covariance functions and the spectral functions.