A linear stochastic differential equation driven by a fractional Brownian motion with Hurst parameter >1/2

TL;DR

This paper investigates solutions to a linear stochastic differential equation driven by fractional Brownian motion with Hurst parameter greater than 1/2, requiring specialized calculus due to the process's non-semimartingale nature.

Contribution

It characterizes the properties of solutions to a linear SDE driven by fractional Brownian motion with Hurst > 1/2, extending stochastic calculus methods for non-semimartingales.

Findings

Solutions exhibit unique properties due to fractional Brownian motion's memory.

A specialized stochastic calculus is developed for non-semimartingale processes.

The paper provides conditions for the existence and uniqueness of solutions.

Abstract

Given a fractional Brownian motion \,\,$(B_{t}^{H})_{t\geq 0}$,\, with Hurst parameter \,$> 1/2$\,\,we study the properties of all solutions of \,\,: {equation} X_{t}=B_{t}^{H}+\int_0^t X_{u}d\mu(u), \;\; 0\leq t\leq 1{equation} A different stochastic calculus is required for the process because it is not a semimartingale.