Spectral functions related to some fractional stochastic differential equations

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.