Sub-fractional $G$-Brownian motion: Properties and simulations

Year 2025, Volume: 54 Issue: 2, 599 - 617, 28.04.2025

Omar Kebiri , Zakaria Boumezbeur , Mhamed Eddahbi , Hacene Boutabia

https://doi.org/10.15672/hujms.1569080

Abstract

In this article, we introduce a new stochastic process called the sub-fractional $G$ -Brownian motion, which serves as an intermediate between the $G$ -Brownian motion and the fractional $G$ -Brownian motion. Although the sub-fractional $G$-Brownian motion shares some properties with the fractional $G$-Brownian motion, it features nonstationary increments. We then examine key characteristics of the process, such as self-similarity, H\"{o}lder continuity, and long-range dependence. Additionally, we propose a method for simulating sample paths of sub-fractional $G$-Brownian motion and conclude by simulating linear stochastic differential equations driven by sub-fractional $G$-Brownian motion.

Keywords

G-Brownian motion , moving average representation , G-expectation , nonstationary increments

Ethical Statement

The first named author acknowledges the funding of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy -The Berlin Mathematics Research Center MATH+(EXC-2046/1, project ID: 390685689), project EF4-6. The second and fourth named authors acknowledge the funding of the ERASMUS KA107 project.The third named author extends his appreciation to the Researchers Supporting Project number (RSPD2025R1075), King Saud University, Riyadh, Saudi Arabia.

Supporting Institution

FU Berlin and King Saud University

Project Number

EXC-2046/1 and RSPD2025R1075

Thanks

The first named author acknowledges the funding of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy -The Berlin Mathematics Research Center MATH+(EXC-2046/1, project ID: 390685689), project EF4-6. The second and fourth named authors acknowledge the funding of the ERASMUS KA107 project.The third named author extends his appreciation to the Researchers Supporting Project number (RSPD2025R1075), King Saud University, Riyadh, Saudi Arabia.

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