Transfer Principle for nth Order Fractional Brownian Motion with Applications to Prediction and Equivalence in Law

TL;DR

This paper develops a transfer principle linking nth order fractional Brownian motion to standard Brownian motion, enabling new prediction formulas and representations for processes equivalent in law, with applications in stochastic modeling.

Contribution

It introduces a novel transfer principle that constructs and represents nth order fractional Brownian motion via Brownian motion, unifying their filtrations and facilitating analysis.

Findings

Constructs a Brownian motion from nth order fractional Brownian motion.

Provides a prediction formula for nth order fractional Brownian motion.

Derives a representation for Gaussian processes equivalent in law.

Abstract

The $n$th order fractional Brownian motion was introduced by Perrin et al. It is the (upto a multiplicative constant) unique self-similar Gaussian process with Hurst index $H \in (n-1,n)$, having $n$th order stationary increments. We provide a transfer principle for the $n$th order fractional Brownian motion, i.e., we construct a Brownian motion from the $n$the order fractional Brownian motion and then represent the $n$the order fractional Brownian motion by using the Brownian motion in a non-anticipative way so that the filtrations of the $n$the order fractional Brownian motion and the associated Brownian motion coincide. By using this transfer principle, we provide the prediction formula for the $n$the order fractional Brownian motion and also a representation formula for all the Gaussian processes that are equivalent in law to the $n$th order fractional Brownian motion.