Work

Fractional and Higher Integer-Order Moments for Fractional Stochastic Differential Equations

Year: 2025

Type: article

Abstract: This study investigates the computation of fractional and higher integer-order moments for a stochastic process governed by a one-dimensional, non-homogeneous linear stochastic differential equation (SDE) driven by fractional Brownian motion (fBm). Unlike conventional approaches relying on moment-generating functions or Fokker–Planck equations, which often yield intractable expressions, we derive explicit closed-form formulas for these moments. Our methodology leverages the Wick–Itô calculus (fractional Itô formula) and the properties of Hermite polynomials to express moments efficiently. Additionally, we establish a recurrence relation for moment computation and propose an alternative approach based on generalized binomial expansions. To validate our findings, Monte Carlo simulations are performed, demonstrating a high degree of accuracy between theoretical and empirical results. The proposed framework provides novel insights into stochastic processes with long-memory properties, with potential applications in statistical inference, mathematical finance, and physical modeling of anomalous diffusion.

Source: Symmetry

DOI: 10.3390/sym17050665

Authors: Arsalane Chouaib Guidoum, Fatimah A. Almulhim, Mohammed Bassoudi, Kamal Boukhetala, Mohammed B. Alamari

Institutions: University of Sciences and Technology Houari Boumediene, University of Tamanghasset, Princess Nourah bint Abdulrahman University, Amar Telidji University of Laghouat, King Khalid University

Language: English

Topic: Stochastic processes and financial applications

Subfield: Finance

Field: Economics, Econometrics and Finance

Domain: Social Sciences

Open Access status: gold

Estimated APC paid: $2,896

Funders: Princess Nourah Bint Abdulrahman University, King Khalid University