高斯过程的局部时和随机流动形
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摘要
近年来,由于分式布朗运动自身自相似性等有趣的特点,被广泛的应用到许多科学领域,因而分式布朗运动的研究成为当前随机分析及其相关领域中的热门之一.但当Hurst参数H≠(?)时,分式布朗运动既不是半鞅,也不是马尔可夫过程.许多随机分析中已有的结论与方法不能直接用于处理分式布朗运动情形.另一方面,将分式布朗运动作为模型有一定的局限性.为了更准确模拟实际情况,迫切地需要我们引入其它的高斯过程.于是,对于这些高斯过程的局部时和随机流动形的研究自然也就成了一个非常有意义且具有挑战性的工作.
在本文中,我们使用白噪声分析方法和Malliavin分析方法来研究高斯过程的局部时和随机流动形.全文的创新性工作如下：
在第3章中我们用白噪声分析方法讨论高斯过程的局部时.首先,证明了关于布朗运动的Wiener积分的广义局部时是一个Hida广义泛函.其次,验证了在给定点处分式布朗运动的局部时是一个Hida广义泛函；利用多重Ito积分给出了局部时的混沌分解.将前面已有结果推广到d维N参数分式布朗运动的局部时情形；利用Hermite多项式得到了局部时的混沌分解.接下来,考虑分式布朗运动的多重相交局部时.在适当的条件下多重相交局部时可以看成一个Hida广义泛函.进一步,将两个相互独立的分式布朗运动的碰撞局部时视为一个Hida广义泛函；在一定的条件下,得到了碰撞局部时的混沌表示与核函数.最后,结合多分式布朗运动的局部非确定性,将分式布朗运动的碰撞局部时推广到两个相互独立的多分式布朗运动情形.
在第4章中我们主要研究高斯过程的随机流动形.首先,我们分别定义Wick积型的布朗随机流动形和分式布朗随机流动形；用白噪声分析方法验证布朗随机流动形和分式布朗随机流动形均为Hida广义泛函.其次,使用Malliavin分析方法,得到双分式布朗随机流动形的正则条件.最后,用类似的方法得到次分式布朗随机流动形的正则条件.
In recent years, fractional Brownian motion (fBm) has become an intense object in stochastic analysis and related fields for the moment, due to its interesting proper-ties, such as self-similarity, and its applications in various scientific areas. However, when Hurst parameter H≠2/1, fBm is neither a semimartingale nor a Markovian pro-cess. FBm can not be directly dealt by many methods and results in stochastic analysis. On the other hand, fBm may be restrictive as a model. In order to simulate the real situation precisely, it is urgent for us to introduce other Gaussian processes. There-fore, it is interesting and challenge work to study local times and stochastic currents of these Gaussian processes.
In this paper, we use white noise analysis approach and Malliavin calculus method to study the local times and stochastic currents of these Gaussian processes. The main innovative results of this paper are as follows.
In section 3, we discuss the local times of Gaussian processes through white noise analysis approach. Firstly, prove that the generalized local time of the Wiener integral with respect to Brownian motion is a Hida distribution. Secondly, for a given point, certify that the local time of fBm is a Hida distribution, and give the chaos expansion of the local time in terms of multiple Ito integral. Similar results of d-dimension fBm with N-parameter are researched. We obtain the chaos expansion of the local time in terms of Hermite polynomial. Thirdly, the multiple intersection local times of fBm are considered. Under the mild conditions, the multiple intersection local times of fBm are regarded as Hida distributions. Fourthly, the collision local times of two inde-pendent fractional Brownian motions are considered as Hida distributions. Under the mild conditions, get the chaos expansions and the kernel functions of two independent fractional Brownian motions. Finally, the results of two independent fractional Brow-nian motions can be extended to the case of two independent multifractional Brownian motions through the similar method and local nondeterministic properties.
In section 4, mainly study the stochastic currents of Gaussian processes. We firstly define Brownian stochastic current and fractional Brownian stochastic current in the sense of Wick integral, respectively. We prove that these stochastic currents are both Hida distributions via white noise analysis method. Next, the conditions of regularity of bifractional Brownian stochastic current are obtained through Malliavin calculus method. Finally, using similar approach, the conditions of regularity of subfractional Brownian stochastic current are obtained.
在本文中,我们使用白噪声分析方法和Malliavin分析方法来研究高斯过程的局部时和随机流动形.全文的创新性工作如下：
在第3章中我们用白噪声分析方法讨论高斯过程的局部时.首先,证明了关于布朗运动的Wiener积分的广义局部时是一个Hida广义泛函.其次,验证了在给定点处分式布朗运动的局部时是一个Hida广义泛函；利用多重Ito积分给出了局部时的混沌分解.将前面已有结果推广到d维N参数分式布朗运动的局部时情形；利用Hermite多项式得到了局部时的混沌分解.接下来,考虑分式布朗运动的多重相交局部时.在适当的条件下多重相交局部时可以看成一个Hida广义泛函.进一步,将两个相互独立的分式布朗运动的碰撞局部时视为一个Hida广义泛函；在一定的条件下,得到了碰撞局部时的混沌表示与核函数.最后,结合多分式布朗运动的局部非确定性,将分式布朗运动的碰撞局部时推广到两个相互独立的多分式布朗运动情形.
在第4章中我们主要研究高斯过程的随机流动形.首先,我们分别定义Wick积型的布朗随机流动形和分式布朗随机流动形；用白噪声分析方法验证布朗随机流动形和分式布朗随机流动形均为Hida广义泛函.其次,使用Malliavin分析方法,得到双分式布朗随机流动形的正则条件.最后,用类似的方法得到次分式布朗随机流动形的正则条件.
In recent years, fractional Brownian motion (fBm) has become an intense object in stochastic analysis and related fields for the moment, due to its interesting proper-ties, such as self-similarity, and its applications in various scientific areas. However, when Hurst parameter H≠2/1, fBm is neither a semimartingale nor a Markovian pro-cess. FBm can not be directly dealt by many methods and results in stochastic analysis. On the other hand, fBm may be restrictive as a model. In order to simulate the real situation precisely, it is urgent for us to introduce other Gaussian processes. There-fore, it is interesting and challenge work to study local times and stochastic currents of these Gaussian processes.
In this paper, we use white noise analysis approach and Malliavin calculus method to study the local times and stochastic currents of these Gaussian processes. The main innovative results of this paper are as follows.
In section 3, we discuss the local times of Gaussian processes through white noise analysis approach. Firstly, prove that the generalized local time of the Wiener integral with respect to Brownian motion is a Hida distribution. Secondly, for a given point, certify that the local time of fBm is a Hida distribution, and give the chaos expansion of the local time in terms of multiple Ito integral. Similar results of d-dimension fBm with N-parameter are researched. We obtain the chaos expansion of the local time in terms of Hermite polynomial. Thirdly, the multiple intersection local times of fBm are considered. Under the mild conditions, the multiple intersection local times of fBm are regarded as Hida distributions. Fourthly, the collision local times of two inde-pendent fractional Brownian motions are considered as Hida distributions. Under the mild conditions, get the chaos expansions and the kernel functions of two independent fractional Brownian motions. Finally, the results of two independent fractional Brow-nian motions can be extended to the case of two independent multifractional Brownian motions through the similar method and local nondeterministic properties.
In section 4, mainly study the stochastic currents of Gaussian processes. We firstly define Brownian stochastic current and fractional Brownian stochastic current in the sense of Wick integral, respectively. We prove that these stochastic currents are both Hida distributions via white noise analysis method. Next, the conditions of regularity of bifractional Brownian stochastic current are obtained through Malliavin calculus method. Finally, using similar approach, the conditions of regularity of subfractional Brownian stochastic current are obtained.
引文
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[2]Mandelbrot B, Van Ness J. Fractional Brownian motions, fractional noises and applications. SI AM Rev.,1968,10:422-437
[3]Flandoli F. On a probabilistic description of small scale structures in 3D fluids. Ann. Inst. H. Poin. Prob. Stat.,2002,38:207-228
[4]Albeverio S, Oliveira M, Streit L. Intersection local times of independent Brownian motions as generalized white noise functionals. Acta Appl. Math.,2001,69:221-241
[5]Boufoussi B, Dozzi M, Guerbaz R. On the local time of multifractional Brownian motion. Stoc: An Inte. J. Prob. Stoc. Proc,2006,78:33-49
[6]Boufoussi B, Dozzi M, Marty R. Local time and Tanaka formula for a Volterra-type multifrac-tional Gaussian process. Bernoulli,2010,16(4):1294-1311
[7]Chen C, Yan L T. Remarks on the intersection local time of fractional Brownian motions. Stat. Prob. Lett., doi:10.1016/j.spl.2011.01.021
[8]Drumond C, Oliveira M, Silva J. Intersection local times of fractional Brownian motions with H (?) (0,1) as generalized white noise functionals.5th Jagna Inte. Workshop Stoc. Quan. Dyna. Biom. Syst.,2008,1021:34-45
[9]Eddahbi M, Lacayo R, Sole J L et al.. Regularity of the local time for the d-dimensional fractional Brownian motion with N-parameters. Stoc. Anal. Appl.,2005,23(2):383-400
[10]Faria M, Hida T, Streit L et al.. Intersection local times as generalized white noise functionals. Acta Appl. Math.,1997,46:351-362
[11]Berman S M. Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J.,1973,23:69-94
[12]Guerbaz R. Holder conditions for the local times of multiscale fractional Brownian motion. C. R. Acad. Sci. Paris Ser.l,2006,343:515-518
[13]Guo J J. Collision local time of two independent multifractional Brownian motions (to appear)
[14]Guo J J, Jiang G, Xiao Y P. Multiple intersection local times of fractional Brownian motion数学杂志,2011,31(3)：388-394
[15]Hu Y Z, Nualart D, Song J. Integral representation of renormalized self-intersection local times. J. Func. Anal.,2008,255:2507-2532
[16]Guo J J, Xiao Y P. Local time of fractional Brownian motion:white noise approach应用数学,2011,24(2)：260-264
[17]Hu Y Z. Self-intersection local time of fractional Brownian motions-via chaos expansion. J. Math. Kyot. Univ.,2001,41:233-250
[18]Hu Y Z,(?)ksendal B. Chaos expansion of local time of fractional Brownian motions. Stoc. Anal. Appl.,2002,20:815-837
[19]Hu Y Z, Nualart D. Regularity of renormalized self-intersection local time for fractional Brownian motion. Comm. Inf. Syst.,2007,7:21-30
[20]Bakun V V. On generalized local time for the process of Brownian motion. Ukra. Math. J.,2000, 52(2):173-182
[21]Imkeller P, Perez-Abreu, Vives J. Chaos expansions of double intersection local time of Brownian motion in Rd and renormalization. Stoc. Proc. Appl.,1995,56:1-34
[22]Imkeller P, Weisz F. The asymptotic behaviour of local times and occupation intergrals of the N-parameter Wiener process in Rd. Prob. Theo. Rela. Fields,1994,98:47-75
[23]Imkeller P, Yan J. Multiple intersection local time of planar Brownian motion as a particular Hida distribution. J. Func. Anal.,1996,140:256-273
[24]Jiang Y M, Wang Y J. On the collision local time of fractional Brownian motions. Chin. Ann. Math.,2007,28:311-320
[25]Wang X J, Guo J J, Jiang G. Collision local times of two independent fractional Brownian motions. Front. Math. China,2011,6(2):325-338
[26]Watanabe H. The local time of self-intersections of Brownian motions as generalized Brownian functionals. Lett. Math. Phys.,1991,23:1-9
[27]Uemura,H. On the weighted local time and the Tanaka formula for the multidimensional fractional Brownian motion. Stoc. Anal. Appl.,2008,26:136-168
[28]Yan L T, Liu J F, Chen C. On the collision local time of bifractional Brownian motions. Stoc. Dyna.,2009,9:479-491
[29]Guo J J. Generalized intersection local time of the indefinite Wiener integral:white noise approach, (to appear)
[30]Yan L T, Shen G J. On the collision local time of sub-fractional Brownian motions. Stat. Prob. Lett.,2010,80:296-308
[31]Marcus M, Rosen J. Additive functionals of several Levy processes and intersection local times. Ann. Prob.,1999,27:1643-1678
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