Skorokhod空间中多参数分数布朗单的一个逼近结果

数学物理学报 ›› 2016, Vol. 36 ›› Issue (1): 168-175.

Skorokhod空间中多参数分数布朗单的一个逼近结果

王志¹, 荆卉婷², 闫理坦³

1 宁波工程学院理学院浙江宁波 315211;
2 宁波工程学院经济管理学院浙江宁波 315211;
3 东华大学数学系上海 201620

收稿日期:2015-07-12 修回日期:2015-12-07 出版日期:2016-02-25 发布日期:2016-02-25
作者简介:王志,wangzhi1006@hotmail.com
基金资助:
国家自然科学基金(11571071,61503205);国家自然科学基金数学天元基金(11526117);浙江省自然科学基金(LQ16A01006);宁波市自然科学基金(2015A610158)和国家级大学生创新创业项目(201511058007)资助

Approximation of Multidimensional Parameter Fractional Brownian Sheet in Skorokhod Space

Wang Zhi¹, Jing Huiting², Yan Litan³

1 School of Sciences, Ningbo University of Technology, Zhejiang Ningbo 315211;
2 School of Economics and Management, Ningbo University of Technology, Zhejiang Ningbo 315211;
3 Department of Mathematics, Donghua University, Shanghai 201620

Received:2015-07-12 Revised:2015-12-07 Online:2016-02-25 Published:2016-02-25
Supported by:
Supported by the NSFC (11571071, 61503205), Mathematical Tianyuan Foundation of China (11526117), Zhejiang Provincial Natural Science Foundation (LQ16A01006), Natural Science Foundation of Ningbo Municipality (2015A610158) and National Students' innovation and entrepreneurship training program (201511058007)

摘要/Abstract

摘要：

证明了一类由独立同分布随机变量构成的随机游走在Skorokhod空间中依分布收敛于多参数分数布朗单.

关键词: 多参数分数布朗单, 随机游走, 弱收敛

Abstract:

In this paper, we prove a convergence in law of a random walk constructed using a family of independent identically distributed random variables to the multidimensional parameter fractional Brownian sheet in the Skorokhod space.

Key words: Multidimensional parameter fractional Brownian sheet, Random walk, Weak convergence

中图分类号:

O211.6

王志, 荆卉婷, 闫理坦. Skorokhod空间中多参数分数布朗单的一个逼近结果[J]. 数学物理学报, 2016, 36(1): 168-175.

Wang Zhi, Jing Huiting, Yan Litan. Approximation of Multidimensional Parameter Fractional Brownian Sheet in Skorokhod Space[J]. Acta mathematica scientia,Series A, 2016, 36(1): 168-175.

参考文献

[1] Bardina X, Florit C. Approximation in law to the d-parameter fractional Brownian sheet based on the functional invariance principle. Rev Mat Iberoamericana, 2005, 21:1037-1052
[2] Bardina X, Jolis M, Tudor C A. Weak convergence to the fractional Brownian sheet and other two-parameter Gaussian processes. Statist Probab Lett, 2003, 65:317-329
[3] Biagini F, Hu Y, Øksendal B, Zhang T. Stochastic Calculus for Fractional Brownian Motion and Applications. London:Springer, 2008
[4] Bickel P J, Wichura M J. Convergence criteria for multiparameter stochastic processes and some applications. Ann Math Statist, 1971, 42:1656-1670
[5] Ciesielski Z, Kamont A. Lévy's fractional Brownian random field and function spaces. Acta Sci Math (Szeged), 1995, 60:99-118
[6] Davydov Y. The invariance principle for stationary processes. Teor Verojatn Primen, 1970, 15:498-509
[7] Decreusefond L, Üstünel A S. Stochastic analysis of the fractional Brownian motion. Potential Anal, 1999, 10:177-214
[8] Delgado R, Jolis M. Weak approximation for a class of Gaussian processes. J Appl Probab, 2000, 37:400-407
[9] Enriquez N. A simple construction of the fractional Brownian motion. Stochastic Process Appl, 2004, 109:203-223
[10] Gradinaru M, Nourdin I, Russo F, Vallois P. m-Order integrals and generalized Itô's formula; the case of a fractional Brownian motion with any Hurst index. Ann Inst H Poincaré Probab Statist, 2005, 41:781-806
[11] Hu Y. Integral Transformations and Anticipative Calculus for Fractional Brownian Motions. New York:Memoirs Amer Math Soc, 2005
[12] Kamont A. On the fractional anisotropic Wiener field. Probab Math Statist, 1996, 16:85-98
[13] Li Y, Dai H. Approximations of fractional Brownian motion. Bernoulli, 2011, 17:1195-1216
[14] Mishura Y S. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Berlin:Springer, 2008
[15] Meyer Y, Sellan F, Taqqu M S. Wavelets, generalized white noise and fractional integration:the synthesis of fractional Brownian motion. J Fourier Anal Appl, 1999, 5:465-494
[16] Nieminen A. Fractional Brownian motion and martingale-differences. Statist Probab Lett, 2004, 70:1-10
[17] Nualart D. Malliavin Calculus and Related Topics. 2nd edition. Berlin:Springer, 2006
[18] Nzi M, Mendy I. Approximation of fractional Brownian sheet by a random walk in anisotropic Besov space. Random Oper Stoch Equ, 2007, 15:137-154
[19] Sottinen T. Fractional Brownian motion, random walks and binary market models. Finance Stoch, 2001, 5:343-355
[20] Taqqu M S. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z Wahrsch Verw Gebiete, 1975, 31:287-302
[21] Tudor C A. Weak convergence to the fractional Brownian sheet in Besov spaces. Bull Braz Math Soc (New Series), 2003, 34:389-400
[22] Wang Z, Yan L, Yu X. Weak approximation of the fractional Brownian sheet from random walks. Electron Commun Probab, 2013, 18:1-13
[23] Wang Z, Yan L, Yu X. Weak convergence to the fractional Brownian sheet using martingale differences. Statist Probab Lett, 2014, 92:72-78