A smooth component of the fractional Brownian motion and optimal portfolio selection

TL;DR

This paper introduces a decomposition of fractional Brownian motion into a smooth and a rough component, enabling the derivation of optimal investment strategies in markets modeled by fractional Brownian motion.

Contribution

It presents a novel representation of fractional Brownian motion as a sum of independent Gaussian processes, including a differentiable-in-mean-square component, and applies this to optimal portfolio selection.

Findings

Decomposition of fractional Brownian motion into smooth and rough parts.

Application to optimal portfolio strategies in fractional Brownian motion markets.

Enhanced understanding of stochastic integrals driven by fractional Brownian motion.

Abstract

We consider fractional Brownian motion with the Hurst parameters from (1/2,1). We found that the increment of a fractional Brownian motion can be represented as the sum of a two independent Gaussian processes one of which is smooth in the sense that it is differentiable in mean square. We consider fractional Brownian motion and stochastic integrals generated by the Riemann sums. As an example of applications, this results is used to find an optimal pre-programmed strategy in the mean-variance setting for a Bachelier type market model driven by a fractional Brownian motion.