Development of a fractional Wiener-Hermite expansion for analyzing the fractional stochastic models

Abstract

The fractional Brownian motion (FBM) is a common model for long and short-range dependent phenomena that appears in different fields, including physics, biology, and finance. In the current work, a new spectral technique named the fractional Wiener Hermite Expansion (FWHE) is developed to analyze stochastic models with FBM. The technique has a theoretical background in the literature and proof of convergence. A new complete orthogonal Hermite basis set is developed. Calculus derivations and statistical analysis are performed to handle the mixed multi-dimensional fractional and/or integer-order integrals that appear in the analysis. Formulas for the mean and variance are deduced and are found to be based on fractional integrals. Using the developed expansion with the statistical properties of the basis functionals will help to reduce the stochastic model to equivalent deterministic fractional models that can be analyzed numerically or analytically with the well-known techniques. A numerical algorithm is developed to be used in case there is no available analytical solution. The numerical algorithm is compared with the fractional Euler-Maruyama (EM) technique to verify the results. In comparison to sampling based techniques, FWHE provides an efficient analytical or numerical alternative. The applicability of FWHE is demonstrated by solving different examples with additive and multiplicative FBM.