带线性自排斥漂移项的分数O-U过程的统计推断
Statistical Inference on the Fractional Ornstein-Uhlenbeck Process with the Linear Self-Repelling Drift
摘要: 本文旨在利用最小二乘法研究带线性自排斥漂移项的分数O-U过程的统计推断。 假设BH=是Hurst 指数为的分数布朗运动，我们考虑下列方程，其中, , θ < 0 和σ, ν ∈ R 是三个参数。 这个过程是自吸引扩散的模拟(见Cranston and Le Jan, Math. Ann. 303 (1995), 87-93)，我们主要的目标是研究其参数的最小二乘估计。
Abstract: This dissertation aim is to study statistical inference on the fractional Ornstein- Uhlenbeck process with the linear self-attracting drift by least squares estimation. Let BH= be a fractional Brownian motion with Hurst index . We consider the following equation, with , where θ < 0 and σ, ν ∈ R are three parameters. The process is an analogue of the self-attracting diffusion (Cranston and Le Jan, Math. Ann. 303 (1995), 87-93). Our main aim is to study the least squares estimations of its parameters.
文章引用：杨晴, 闫理坦. 带线性自排斥漂移项的分数O-U过程的统计推断[J]. 应用数学进展, 2023, 12(5): 2235-2254. https://doi.org/10.12677/AAM.2023.125229

参考文献

[1] Durrett, R. and Rogers, L.C.G. (1991) Asymptotic Behavior of Brownian Polymer. Probability Theory and Related Fields, 92, 337-349.
https://doi.org/10.1007/BF01300560
[2] Coppersmith, D. and Diaconis, P. (1986) Random Walks with Reinforcement. Unpublished manuscript.
[3] Cranston, M. and Le Jan, Y. (1995) Self-Attracting Diffusions: Two Case Studies. Mathema- tische Annalen, 303, 87-93.
https://doi.org/10.1007/BF01460980
[4] % Bena¨ım, M., Ciotir, I. and Gauthier, C.-E. (2015) Self-Repelling Diffusions via an Infinite Dimensional Approach. Stochastic Partial Differential Equations: Analysis and Computations, 3, 506-530.
https://doi.org/10.1007/s40072-015-0059-5
[5] Gauthier, C.-E. (2016) Self Attracting Diffusions on a Sphere and Application to a Periodic Case. Electronic Communications in Probability, 21, 1-12.
https://doi.org/10.1214/16-ECP4547
[6] Herrmann, S. and Roynette, B. (2003) Boundedness and Convergence of Some Self-Attracting Diffusions. Mathematische Annalen, 325, 81-96.
https://doi.org/10.1007/s00208-002-0370-0
[7] Herrmann, S. and Scheutzow, M. (2004) Rate of Convergence of Some Self-Attracting Diffu- sions. Stochastic Processes and Their Applications, 111, 41-55.
https://doi.org/10.1016/j.spa.2003.10.012
[8] Bena¨ım, M., Ledoux, M. and Raimond, O. (2002) Self-Interacting Diffusions. Probability The- ory and Related Fields, 122, 1-41.
https://doi.org/10.1007/s004400100161
[9] Chambeu, S. and Kurtzmann, A. (2011) Some Particular Self-Interacting Diffusions: Ergodic Behaviour and Almost Sure Convergence. Bernoulli, 17, 1248-1267.
https://doi.org/10.3150/10-BEJ310
[10] Cranston, M. and Mountford, T.S. (1996) The Strong Law of Large Numbers for a Brownian Polymer. Annals of Probability, 24, 1300-1323.
https://doi.org/10.1214/aop/1065725183
[11] Mountford, T. and Tarr`es, P. (2008) An Asymptotic Result for Brownian Polymers. Annales de l’Institut Henri Poincar´e, Probabilit´es et Statistiques, 44, 29-46.
https://doi.org/10.1214/07-AIHP113
[12] Yan, L., Sun, Y. and Lu, Y. (2008) On the Linear Fractional Self-Attracting Diffusion. Journal of Theoretical Probability, 21, 502-516.
https://doi.org/10.1007/s10959-007-0113-y
[13] Sun, X., Yan, L. and Ge, Y. (2022) The Laws of Large Numbers Associated with the Linear Self-Attracting Diffusion Driven by Fractional Brownian Motion and Applications. Journal of Theoretical Probability, 35, 1423-1478.
https://doi.org/10.1007/s10959-021-01126-0
[14] Chakravari, N. and Sebastian, K. (1997) Fractional Brownian Motion Models for Polymers. Chemical Physics Letters, 267, 9-13.
https://doi.org/10.1016/S0009-2614(97)00075-4
[15] Cherayil, B. and Biswas, P. (1993) Path Integral Description of Polymers Using Fractional Brownian Walks. The Journal of Chemical Physics, 99, 9230-9236.
https://doi.org/10.1063/1.465539
[16] Yan, L., Yang, Q. and Xia, X. Long Time Behavior on the Fractional Ornstein-Uhlenbeck Process with the Linear Self-Repelling Drift.
[17] Biagini, F., Hu, Y., Øksendal, B. and Zhang, T. (2008) Stochastic Calculus for Fractional Brownian Motion and Applications. In: Probability and Its Application, Springer, Berlin.
[18] Hu, Y. (2005) Integral Transformations and Anticipative Calculus for Fractional Brownian Mo- tions. In: Memoirs of the American Mathematical Society, Vol. 175, American Mathematical Society, Rhode Island.
https://doi.org/10.1090/memo/0825
[19] Mishura, Y.S. (2008) Stochastic Calculus for Fractional Brownian Motion and Related Pro- cesses. In: Lecture Notes in Mathematics, Vol. 1929, Springer, Berlin, Heidelberg.
https://doi.org/10.1007/978-3-540-75873-0
[20] Nualart, D. (2006) Malliavin Calculus and Related Topics. 2nd Edition, Springer, Heidelberg, New York.
[21] Nourdin, I. (2012) Selected Aspects of Fractional Brownian Motion. Springer-Verlag, Milano.
https://doi.org/10.1007/978-88-470-2823-4
[22] Tudor, C. (2013) Analysis of Variations for Self-Similar Processes. Springer, Heidelberg, New York.
[23] Nualart, D. and Ortiz-Latorre, S. (2008) Central Limit Theorems for Multiple Stochastic integrals And Malliavin Calculus. Stochastic Processes and Their Applications, 118, 614-628.
https://doi.org/10.1016/j.spa.2007.05.004
[24] Nualart, D. and Peccati, G. (2005) Central Limit Theorems for Sequences of Multiple Stochas- tic Integrals. Annals of Probability, 33, 177-193.
https://doi.org/10.1214/009117904000000621
[25] Decreusefond, L. and U¨ stu¨nel, A.S. (1999) Stochastic Analysis of the Fractional Brownian Motion. Potential Analysis, 10, 177-214.
https://doi.org/10.1023/A:1008634027843
[26] Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Langhorne, PA.
[27] Es-Sebaiy, K. and Nourdin, I. (1991) Parameter Estimation for α Fractional Bridges. Proba- bility Theory and Related Fields, 92, 337-349.

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