Evaluation of integrals with fractional Brownian motion for different Hurst indices

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. (Formula presented.). Particularly, the fractional Ornstein–Uhlenbeck (fOU) process 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 (Formula presented.) to the larger domain (Formula presented.). Numerical experiments for different Hurst indices confirm the robustness and efficiency of the integral formulations presented. 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.