Araştırma Makalesi
BibTex RIS Kaynak Göster

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

Yıl 2025, Cilt: 54 Sayı: 2, 599 - 617, 28.04.2025

Omar Kebiri , Zakaria Boumezbeur , Mhamed Eddahbi , Hacene Boutabia

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

Öz

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.

Anahtar Kelimeler

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

Etik Beyan

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.

Destekleyen Kurum

FU Berlin and King Saud University

Proje Numarası

EXC-2046/1 and RSPD2025R1075

Teşekkür

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.

Kaynakça

  • [1] M. Aneta and F. Darya, On the simulation of sub-fractional Brownian motion, 20th International Conference on Methods and Models in Automation and Robotics (MMAR), 400-405, 2015.
  • [2] T. Bojdecki, L. Gorostiza and A. Talarczyk, Sub-fractional Brownian motion and its relation to occupation times, Stat. Probab. Lett. 69 (4), 405-419, 2004.
  • [3] T.T. Dufera, Fractional Brownian motion in option pricing and dynamic delta hedging: Experimental simulations, N. Am. J. Econ. Finance. 69, Part B, 2024.
  • [4] M.D.N. da Costa, A.A. Brito, N.P.A. Castro, S.T.M.R. Dias and F.G. Zebende, Trends in the Air Temperature: A Practical Approach for Auto-and Cross-Correlation Analysis, Adv. Meteorol. 2024 (1), 2024.
  • [5] D. Feyel and A. De La Pradelle, On fractional Brownian processes, Potential Anal. 10, 273-288, 1999.
  • [6] W. Hu, Q. Yang, L. Peng, L. Liu, P. Zhang, S. Li and J. Wu, Non-stationary modeling and simulation of strong winds, Heliyon 10 (15), 2024.
  • [7] Y. Jicheng, F. Yuqiang and W. Xianjia, Lie symmetry, exact solutions and conservation laws of bi-fractional BlackScholes equation derived by the fractional G-Brownian motion, Int. J. Financ. Eng. 11 (1), 2024.
  • [8] A. E. Kyojo, E. S. Osima, S. S. Mirau and G. V. Masanja, Applying Stationary and Nonstationary Generalized Extreme Value Distributions in Modeling Annual Extreme Temperature Patterns, Adv. Meteorol. 2024 (1), 2024.
  • [9] D. Laurent, M. Hu and S. Peng, Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion paths, Potential Anal. 34, 139-161, 2011.
  • [10] R. Monjo and O. Meseguer-Ruiz, Review: Fractal Geometry in Precipitation, Atmosphere 15 (1), 2024.
  • [11] S. Peng, Stochastic Analysis and Applications: The Abel Symposium 2005, Springer Berlin, Heidelberg, 2007.
  • [12] S. Peng, G-Brownian motion and dynamic risk measure under volatility uncertainty, arXiv:0711.2834 [math.PR].
  • [13] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty: with robust CLT and G-Brownian motion, Springer Nature, 2019.
  • [14] F. Shokrollahi, D. Ahmadian and L.V. Ballestra, Pricing Asian options under the mixed fractional Brownian motion with jumps, Math. Comput. Simul. 226, 172-183, 2024.
  • [15] C. Tudor, Some properties of the sub-fractional Brownian motion, Stochastics 79 (5), 431-448, 2007.
  • [16] A.C. Tudor, Analysis of variations for self-similar processes: a stochastic calculus approach, Springer Science & Business Media, 2013.
  • [17] C. Wei, G-white noise theory, wavelet decomposition for fractional G-Brownian motion, and bid-ask pricing application to finance under uncertainty, arXiv:1306.4070 [q-fin.PR].
  • [18] J. Yang and W. Zhao, Numerical simulations for G-Brownian motion, Front. Math. China 11, 1625-1643, 2016.
  • [19] L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1), 251-282, 1936.

Yıl 2025, Cilt: 54 Sayı: 2, 599 - 617, 28.04.2025

Omar Kebiri , Zakaria Boumezbeur , Mhamed Eddahbi , Hacene Boutabia

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

Öz

Proje Numarası

EXC-2046/1 and RSPD2025R1075

Kaynakça

  • [1] M. Aneta and F. Darya, On the simulation of sub-fractional Brownian motion, 20th International Conference on Methods and Models in Automation and Robotics (MMAR), 400-405, 2015.
  • [2] T. Bojdecki, L. Gorostiza and A. Talarczyk, Sub-fractional Brownian motion and its relation to occupation times, Stat. Probab. Lett. 69 (4), 405-419, 2004.
  • [3] T.T. Dufera, Fractional Brownian motion in option pricing and dynamic delta hedging: Experimental simulations, N. Am. J. Econ. Finance. 69, Part B, 2024.
  • [4] M.D.N. da Costa, A.A. Brito, N.P.A. Castro, S.T.M.R. Dias and F.G. Zebende, Trends in the Air Temperature: A Practical Approach for Auto-and Cross-Correlation Analysis, Adv. Meteorol. 2024 (1), 2024.
  • [5] D. Feyel and A. De La Pradelle, On fractional Brownian processes, Potential Anal. 10, 273-288, 1999.
  • [6] W. Hu, Q. Yang, L. Peng, L. Liu, P. Zhang, S. Li and J. Wu, Non-stationary modeling and simulation of strong winds, Heliyon 10 (15), 2024.
  • [7] Y. Jicheng, F. Yuqiang and W. Xianjia, Lie symmetry, exact solutions and conservation laws of bi-fractional BlackScholes equation derived by the fractional G-Brownian motion, Int. J. Financ. Eng. 11 (1), 2024.
  • [8] A. E. Kyojo, E. S. Osima, S. S. Mirau and G. V. Masanja, Applying Stationary and Nonstationary Generalized Extreme Value Distributions in Modeling Annual Extreme Temperature Patterns, Adv. Meteorol. 2024 (1), 2024.
  • [9] D. Laurent, M. Hu and S. Peng, Function spaces and capacity related to a sublinear expectation: Application to G-Brownian motion paths, Potential Anal. 34, 139-161, 2011.
  • [10] R. Monjo and O. Meseguer-Ruiz, Review: Fractal Geometry in Precipitation, Atmosphere 15 (1), 2024.
  • [11] S. Peng, Stochastic Analysis and Applications: The Abel Symposium 2005, Springer Berlin, Heidelberg, 2007.
  • [12] S. Peng, G-Brownian motion and dynamic risk measure under volatility uncertainty, arXiv:0711.2834 [math.PR].
  • [13] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty: with robust CLT and G-Brownian motion, Springer Nature, 2019.
  • [14] F. Shokrollahi, D. Ahmadian and L.V. Ballestra, Pricing Asian options under the mixed fractional Brownian motion with jumps, Math. Comput. Simul. 226, 172-183, 2024.
  • [15] C. Tudor, Some properties of the sub-fractional Brownian motion, Stochastics 79 (5), 431-448, 2007.
  • [16] A.C. Tudor, Analysis of variations for self-similar processes: a stochastic calculus approach, Springer Science & Business Media, 2013.
  • [17] C. Wei, G-white noise theory, wavelet decomposition for fractional G-Brownian motion, and bid-ask pricing application to finance under uncertainty, arXiv:1306.4070 [q-fin.PR].
  • [18] J. Yang and W. Zhao, Numerical simulations for G-Brownian motion, Front. Math. China 11, 1625-1643, 2016.
  • [19] L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1), 251-282, 1936.
Toplam 19 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Olasılıksal Analiz ve Modelleme
Bölüm İstatistik
Yazarlar

Omar Kebiri 0000-0002-1420-2071

Zakaria Boumezbeur 0000-0002-7654-2514

Mhamed Eddahbi 0000-0003-0889-3387

Hacene Boutabia 0000-0003-1319-2091

Proje Numarası EXC-2046/1 and RSPD2025R1075
Erken Görünüm Tarihi 31 Ocak 2025
Yayımlanma Tarihi 28 Nisan 2025
Gönderilme Tarihi 17 Ekim 2024
Kabul Tarihi 16 Ocak 2025
Yayımlandığı Sayı Yıl 2025 Cilt: 54 Sayı: 2

Kaynak Göster

APA Kebiri, O., Boumezbeur, Z., Eddahbi, M., Boutabia, H. (2025). Sub-fractional $G$-Brownian motion: Properties and simulations. Hacettepe Journal of Mathematics and Statistics, 54(2), 599-617. https://doi.org/10.15672/hujms.1569080
AMA Kebiri O, Boumezbeur Z, Eddahbi M, Boutabia H. Sub-fractional $G$-Brownian motion: Properties and simulations. Hacettepe Journal of Mathematics and Statistics. Nisan 2025;54(2):599-617. doi:10.15672/hujms.1569080
Chicago Kebiri, Omar, Zakaria Boumezbeur, Mhamed Eddahbi, ve Hacene Boutabia. “Sub-Fractional $G$-Brownian Motion: Properties and Simulations”. Hacettepe Journal of Mathematics and Statistics 54, sy. 2 (Nisan 2025): 599-617. https://doi.org/10.15672/hujms.1569080.
EndNote Kebiri O, Boumezbeur Z, Eddahbi M, Boutabia H (01 Nisan 2025) Sub-fractional $G$-Brownian motion: Properties and simulations. Hacettepe Journal of Mathematics and Statistics 54 2 599–617.
IEEE O. Kebiri, Z. Boumezbeur, M. Eddahbi, ve H. Boutabia, “Sub-fractional $G$-Brownian motion: Properties and simulations”, Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 2, ss. 599–617, 2025, doi: 10.15672/hujms.1569080.
ISNAD Kebiri, Omar vd. “Sub-Fractional $G$-Brownian Motion: Properties and Simulations”. Hacettepe Journal of Mathematics and Statistics 54/2 (Nisan 2025), 599-617. https://doi.org/10.15672/hujms.1569080.
JAMA Kebiri O, Boumezbeur Z, Eddahbi M, Boutabia H. Sub-fractional $G$-Brownian motion: Properties and simulations. Hacettepe Journal of Mathematics and Statistics. 2025;54:599–617.
MLA Kebiri, Omar vd. “Sub-Fractional $G$-Brownian Motion: Properties and Simulations”. Hacettepe Journal of Mathematics and Statistics, c. 54, sy. 2, 2025, ss. 599-17, doi:10.15672/hujms.1569080.
Vancouver Kebiri O, Boumezbeur Z, Eddahbi M, Boutabia H. Sub-fractional $G$-Brownian motion: Properties and simulations. Hacettepe Journal of Mathematics and Statistics. 2025;54(2):599-617.

For more information about the journal, please visit: https://dergipark.org.tr/en/pub/hujms