Parameter Estimation for Brownian Motion with Trend Mixed with Two Fractional Brownian Motions
Article Sidebar
Main Article Content
Abstract
This work studies parameter estimation for a stochastic process that combines a Brownian motion with trend and two fractional Brownian motions with distinct Hurst indices. The process is given by , where
is a standard Brownian motion and
are fractional Brownian motions with Hurst indices
such that
, with
,
and
being mutually independent. An ergodic theorem is applied to derive estimators for all unknown parameters and examine the effect of different Borel function selections on the estimator formulations. Numerical simulations are performed to confirm the asymptotic properties of the estimators and to assess their accuracy under various configurations. The estimators are then applied to cryptocurrency data to demonstrate their practical use.
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
References
Kolmogorov AN. Wienersche Spiralen und einige andere interessante Kurven im Raum. C R (Doklady) Acad Sci URSS (NS) 1940;26:115-8.
Biagini F, Hu Y, Øksendal B, Zhang T. Stochastic calculus for fractional Brownian motion and applications. London: Springer; 2008.
Mishura YS. Stochastic calculus for frac-tional Brownian motion and related processes. Berlin: Springer; 2008.
Rostek S, Schöbel R. A note on the use of fractional Brownian motion for financial modeling. Econ Model 2013;30:30-5.
Decreusefond L, Üstünel AS. Fractional Brownian motion: theory and applica-tions. ESAIM Proc 1998;5:75-86.
Ibrahim SNI, Misiran M, Laham MF.Geometric fractional Brownian motion model for commodity market simulation. Alex Eng J 2021;60:955-62.
Kubilius K, Skorniakov V, Melichov D. Estimation of parameters of SDE driven by fractional Brownian motion with polynomial drift. J Stat ComputSimul 2016;86:1954-69.
Cheridito P. Mixed fractional Brownian motion. Bernoulli 2001;7:913-34.
Filatova D. Mixed fractional Brownian motion: some related questions for computer network traffic modeling. In: Int Conf Signals Electron Syst; 2008. p. 393-6.
Sun L. Pricing currency options in the mixed fractional Brownian motion. Phys A: Stat Mech Appl 2013;392:3441-58.
Zhang WG, Xiao WL, He CX. Equity warrants pricing model under fractional Brownian motion and an empirical study. Expert Syst Appl 2009;36:3056-65.
Xiao WL, Zhang WG, Zhang X. Maximum-likelihood estimators in the mixed fractional Brownian motion. Statistics 2011;45:73-85.
Zhang P, Sun Q, Xiao WL. Parameter identification in mixed Brownianfractional Brownian motions using Powell’s optimization algorithm. Econ Model 2014;40:314-9.
Dozzi M, Mishura Y, Shevchenko G. Asymptotic behavior of mixed power variations and statistical estimation in mixed models. Stat Inference Stoch Process 2015;18:151-75.
Dufitinema J, Pynnönen S, Sottinen T. Maximum likelihood estimators from discrete data modeled by mixed fractional Brownian motion with application to the Nordic stock markets. Commun Statist Simulation Comput 2022;51:5264-87.
Kukush A, Lohvinenko S, Mishura Y, Ralchenko K. Two approaches to consistent estimation of parameters of mixed fractional Brownian motion with trend. Stat Inference Stoch Process 2022;25:159-87.
Miao Y, Ren W, Ren Z. On the fractional mixed fractional Brownian motion. Appl Math Sci (Ruse) 2008;2:1729-38.
Mishura Y. Maximum likelihood drift estimation for the mixing of two fractional Brownian motions. In: Stochastic and Infinite Dimensional Analysis (Trends in Mathematics). Cham: /Springer; 2016. p. 263-80.
Mishura Y, Voronov I. Construction of maximum likelihood estimator in the mixed fractional-fractional Brownian motion model with double long-range dependence. Mod Stoch Theory Appl 2015;2:147-64.
Ralchenko K, Yakovliev M. Parameter estimation for fractional mixed fractional Brownian motion based on discrete observations. Mod Stoch Theory Appl 2024;11:1-29.
Higham DJ. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev 2001;43:525-46.
Shevchenko G. Fractional Brownian motion in a nutshell. Int J Mod Phys Conf Ser 2015;36:1560002.
Wood ATA, Chan G. Simulation of stationary Gaussian processes in [0, 1]𝑑 . J Comput Graph Statist 1994;3:409-32.
Maruyama G. The harmonic analysis of stationary stochastic processes. Memoirs of the Faculty of Science, Kyushu University 1949;4:45-106.
Bennedsen M, Christensen K, Christensen P. To be or not to be: Roughness or long memory in volatility? J Econometrics 2026;254:106193.
