Solution of integrals with fractional Brownian motion for different Hurst indices

TL;DR

This paper develops methods to evaluate integrals related to fractional Brownian motion for all Hurst indices, extending classical techniques and confirming results through numerical experiments and applications in financial option pricing.

Contribution

It introduces an analytic continuation approach to extend integral formulas for fractional Brownian motion across all Hurst indices, enhancing theoretical understanding and computational efficiency.

Findings

Integral formulas valid for all H in (0,1)

Numerical experiments confirm robustness of methods

Efficient option pricing using Fourier cosine expansions

Abstract

In this paper, we will evaluate integrals that define the conditional expectation, variance and characteristic function of stochastic processes with respect to fractional Brownian motion (fBm) for all relevant Hurst indices, i.e. $H \in (0,1)$. The fractional Ornstein-Uhlenbeck (fOU) process, for example, gives rise to highly nontrivial integration formulas that need careful analysis when considering the whole range of Hurst indices. We will show that the classical technique of analytic continuation, from complex analysis, provides a way of extending the domain of validity of an integral, from $H\in(1/2,1)$, to the larger domain, $H\in(0,1)$. Numerical experiments for different Hurst indices confirm the robustness and efficiency of the integral formulations presented here. Moreover, we provide accurate and highly efficient financial option pricing results for processes that are related to the fOU process, with the help of Fourier cosine expansions.