Fractional Brownian Motion and the Markov Property

Abstract

Fractional Brownian motion belongs to a class of long memory Gaussian processes that can be represented as linear functionals of an infinite dimensional Markov process. This leads naturally to:

• An efficient algorithm to approximate the process.

• An ergodic theorem which applies to functionals of the type $$\int_0^t \phi(V_h(s)),ds \quad\text{where}\quad V_h(s)=\int_0^s h(s-u), dB_u,.$$

where $B$ is a real Brownian motion.