Gel'fand三元组上的Lévy白噪声和分数Lévy噪声
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摘要
分数Brown运动的长相依，自相似性质使得它在金融，经济，网络通讯等领域有着广泛的应用，引起了人们的广泛关注。分数Brown运动可表示为由Riemann-Liouville分数积分算子确定的Volterra型核函数关于布朗运动的积分，借助此积分核函数关于0均值平方可积L(?)vy过程的积分可以构造更一般的分数L(?)vy过程，因而大大推广了分数过程的应用范围。
另一方面，由分数噪声驱动的偏微分方程的研究，需要构造无穷维向量空间上的分数过程，但是，迄今为止，只限于Hilbert空间上由分数Brown运动驱动的随机发展方程的研究。我们知道，取值于Hilbert空间V上的分数Brown运动存在的必要充分条件是它的协方差算子为核算子，否则，它将取值于一个更大的Hilbert空间V_1使得V连续稠密的嵌入V_1且嵌入映射为Hilbert-Schmidt的。很自然地，我们考虑取值于可列Hilbert核空间的对偶空间上的分数过程，最合适的空间结构就是Gel'fand三元组。
令H为实可分Hilbert空间，|·|_0和〈．，．〉分别为其上的范数和内积。A为H上正的自伴算子，且存在α＞0使得A~(-α)为核算子。对(?)r∈R，定义|·|_r：=|A~r·|_0，令E_r为A~r的定义域关于|·|_r的完备化，则E_r为实可分Hilbert空间，E_r和E_(-r)互为对偶空间。令E为{E_r}_r(?)0的投影极限，E~*为{E_(-r)}_r＞0的归纳极限，则E为可列Hilbert核空间且E~*为其对偶。记〈·，·〉为E~*×E上的典则双线性型，(?)~+≡(?)~+(E，E~*)为E到E~*上的正的连续线性算子，E(?)H(?)E~*称为由(H，A)生成实Gel'fand三元组。本文采用这样的三元组作为基本框架是因为：
1°它包含了常见的许多无穷维向量空间，如Schwartz缓增广义函数空间、Hida白噪声广义泛函空间等，也是有限维欧氏空间的自然推广；
2°其核空间结构使我们可以灵活地使用拓扑张量积、Bochner-Minlos定理、Schwartz核定理、It(?)正则化定理等，克服构造中的一系列困难。
以下是本文的几个主要结论：
结论一令E (?)H(?) E~*为由(H，A)生成实Gel'fand三元组。对于给定α∈E~*，Q∈(?)+和E~*上的测度v，其中v的支撑为E_(-p)且满足则存在q＞0和E~*上的无穷可分分布μ，使得我们称(α，Q，v)为μ的生成三元组，v为L(?)vy测度。
这样，我们得到了E~*上最一般的无穷可分分布特征泛函的表达式，特别地，我们构造了E~*上的稳定分布。然后，我们定义了E~*-值的L(?)vy过程，并给出其L(?)vy-It(?)分解。利用再生核Hilbert空间技巧，我们构造了一类特殊的算子值过程关于平方可积的L(?)vy过程的随机积分。
结论二基于E~*上无穷可分分布特征泛函的表达式，我们定义了E~*-值的L(?)vy白噪声。设X={X_t，t∈R}为(α，Q，v)生成的E~*-值L(?)vy过程，其中α∈E~*，Q∈(?)~+，L(?)vy测度v满足∫_(|x|_(-p)＞1)|x|_(-p)dv(x)＜∞，X_1的特征指数ψ由结论一给出。则对(?)f∈L~1(R)∩L~2(R)，为E~*-值R．V．使得则{(?)(f)，f∈S(R)}为E~*-值缓增白噪声，我们称其为E~*-值的L(?)vy白噪声。
结论三由结论二以及Riemann-Liouville分数积分算子I~β_的连续性，我们构造了一个新的E~*-值(缓增)广义过程作为E~*-值的L(?)vy白噪声的泛函，称为E~*-值的分数L(?)vy噪声。在结论二的条件下，定义则{(?)~β(f)，f∈S(R)}为E~*-值(缓增)广义过程(我们称为E~*-值β-分数L(?)vy噪声)。而且，特别地，对示性函数f=1_([0，t])，只要下面积分存在，我们就可以定义如下的E~*-值β-分数L(?)vy过程{X_t~β，t(?)0}：
结论四特别地，当E~*-值的L(?)vy过程X平方可积时，作为缓增的L(?)vy白噪声的泛函，我们研究了E~*-值的分数L(?)vy过程{X_t~β：=(?)~β(1_([0，t]))，t(?)0}的样本轨道性质和分布性质，证明了分数L(?)vy过程具有平稳增量性，长相依性质，对一类特殊的算子值过程讨论关于分数L(?)vy过程的随机积分，给出分数L(?)vy过程的新息表示。并将E~*-值分数L(?)vy过程作了推广，定义了组合分数L(?)vy过程和多分数L(?)vy过程。
结论五上述方法和结论可特别方便地推广到多维时间参数情形，在E~*上无穷可分分布的一阶矩存在的条件下，我们构造了E~*-值L(?)vy随机场。并利用Riesz位势，Riesz多重位势以及多变量型的Riemann-Liouville分数积分算子，构造了Gel，fand三元组上各向同性分数L(?)vy随机场，各向异性分数L(?)vy随机场以及多参数分数L(?)vy过程，并分别研究了它们的性质(Euclid不变性，长相依性，自相似性，平稳增量性等)。
By virtue of its self-similarity and long-range dependence,fractional Brownian mo-tion suits to model driving noises in different applications such as mathematical finance,economics,and network traffic analysis,and has evoked wide interests.Fractional Brown-ian motion can be represented as stochastic integral of a kernel function of Volterra typedetermined by the Riemann-Liouville fractional integral and derivative operator with re-spect to Brownian motion.By stochastic integration of this kernel function with respectto a general L(?)vy process,one can define fractional L(?)vy process which greatly widenthe fields of applications of fractional processes.
On the other hand,for the study on stochastic differential equations driven byfractional noises in infinitely dimensional vector spaces,one need to constuct fractionalprocesses in these spaces.So far as we know,the current study only limited to stochasticevolution equations driven by fractional Brownian motions in Hilbert spaces.However,the fractional Brownian motion B~H in a Hilbert space V exists if and only if its covarianceoperator Q is nuclear.Otherwise,it will take values in a larger Hilbert space V_1 suchthat V continuously imbedded into V_1 and the imbedding is a Hilbert-Schmidt operator.So,it is more natural to consider the fractional processes taking values in the dual spaceof a countably Hilbertian nuclear space.The most suitable framework should be theGel'fand triple.
Let H be a real separable Hilbert space with norm |·|_0 and inner product〈·,·〉.LetA be a positive self-adjoint operator in H such that A~(-α) is nuclear for someα＞0.Foreach r∈R,define |·|_r:=|A~r·|_0 and let E_r be the completion of the domain of A~rwith respect to |·|_r.Then E_r is a real Hilbert space,E_r and E_(-r) can be viewed as eachother's duals.Let E be the projective limit of {E_r}_r≥0 and E~* be the inductive limit of{E_(-r)}_r≥0.Then E is a Fr(?)chet nuclear space with E~* as its dual.We denote by〈·,·〉the canonical bilinear form on E~*×E,(?)~+≡(?)~+(E,E~*) the space of positive continuouslinear maps from E to E~*,then E(?)H(?)E~* is referred to as a Gel'fand triple generatedby (H,A).All the work of this dissertation is under this framework since
1°It includs many usual infinitely dimensional spaces such as Schwartz space of tempered distributions,Hida space of generalized white noise functionals,and is themost natural extension of finite dimensional Euclidean spaces;
2°Under this framework,we can freely use topological tensor product,Bochner-Minlos theorem,Schwartz kernel theorem,It(?) regularization theorem etc.to overcomemany difficulty in the construction.
The main conclusions of this dissertation are as following:
Conclusion 1 Let E(?)H(?)E~* be a real Gel'fand triple generated by (H,A) asabove.Given a∈E~*,Q∈(?)~+ and a Borel measure v on E~* supported in E_(-p) for somep＞0 satisfyingThen there exists q＞0 and an infinitely divisible distributionμon E~* such thatWe call (a,Q,v) the generating triple ofμand v the L(?)vy measure.
So,we obtain a general form of the charateristic functional of infinitely divisibledistributions on E~*,as a special case,we construct stable distributions on E~*.Thenwe define the E~*-valued L(?)vy process and give its L(?)vy-It(?) decomposition.Using areproducing kernel Hilbert space technique,we define stochastic integrals for a class ofoperator-valued processes with respect to L(?)vy processes.
Conclusion 2 By virtue of the charateristic functional of the infinitely divisibledistribution on E~*,we introduce E~*-valued L(?)vy white noises.Let X={X_t,t∈R} bean E~*-valued L(?)vy process generated by triple (a,Q,v) with a∈E~*,Q∈(?)~+ and L(?)vymeasure v satisfying f_(|x|-p＞1)|x|_(-p)dv(x)＜∞whose characteristic exponent is given byConclusion 1.Then,for any f∈L~1(R)∩L~2(R),is a well-defined E~*-valued R.V.such that Therefore,{(?)(f),f∈S(R)} is a tempered E~*-valued white noise.We call it L(?)vy whitenoise on Gel'fand triple E(?)H(?)E~*.
Conclusion 3 By Conclusion 2 and the continuity of the Riemann-Liouville frac-tional integration operator I_-~β,we construct a new E~*-valued (tempered) generalizedprocess as functional of E~*-valued L(?)vy white noise which is referred to as E~*-valuedfractional L(?)vy noise.Under the conditions of Conclusion 2,defineThen {(?)~β(f),f∈S(R)} is an E~*-valued (tempered) generalized process (referred to asE~*-valuedβ-fractional L(?)vy noise).Moreover,For f=1_([0,t]),we obtain the E~*-valuedβ-fractional L(?)vy process:whenever the right hand side exists.
Conclusion 4 When the underlying L(?)vy process is square integrable,we constructthe E~*-valuedβ-fractional L(?)vy process {X_t~β:= (?)~β(1_([0,t])),t≥0} and investigate itsdistribution and sample properties by considering it as functional of the E~*-valued L(?)vywhite noise.We prove that fractional L(?)vy process has stationary increments,long-range dependence,and give its innovational representation.In addition,we extend theconstruction to mixed fractional L(?)vy process and multi-fractional L(?)vy process.
Conclusion 5 The above ideas and conclusions could be easily generalized to thecase of higher dimensions of the time parameter.Under the condition that the firstmoment of the infinitely divisible distribution on E~* exists,replacing the operator I_-~βby Riesz potential,Riesz poly-potential and multi-variate fractional integration opera-tor respectively,we constuct the E~*-valued L(?)vy random field,the isotropic fractionalL(?)vy random fields and anisotropic fractional L(?)vy random fields,and investigate theirdistribution and sample properties such as Euclidean invariance,long-range dependence,self-similarity.
另一方面，由分数噪声驱动的偏微分方程的研究，需要构造无穷维向量空间上的分数过程，但是，迄今为止，只限于Hilbert空间上由分数Brown运动驱动的随机发展方程的研究。我们知道，取值于Hilbert空间V上的分数Brown运动存在的必要充分条件是它的协方差算子为核算子，否则，它将取值于一个更大的Hilbert空间V_1使得V连续稠密的嵌入V_1且嵌入映射为Hilbert-Schmidt的。很自然地，我们考虑取值于可列Hilbert核空间的对偶空间上的分数过程，最合适的空间结构就是Gel'fand三元组。
令H为实可分Hilbert空间，|·|_0和〈．，．〉分别为其上的范数和内积。A为H上正的自伴算子，且存在α＞0使得A~(-α)为核算子。对(?)r∈R，定义|·|_r：=|A~r·|_0，令E_r为A~r的定义域关于|·|_r的完备化，则E_r为实可分Hilbert空间，E_r和E_(-r)互为对偶空间。令E为{E_r}_r(?)0的投影极限，E~*为{E_(-r)}_r＞0的归纳极限，则E为可列Hilbert核空间且E~*为其对偶。记〈·，·〉为E~*×E上的典则双线性型，(?)~+≡(?)~+(E，E~*)为E到E~*上的正的连续线性算子，E(?)H(?)E~*称为由(H，A)生成实Gel'fand三元组。本文采用这样的三元组作为基本框架是因为：
1°它包含了常见的许多无穷维向量空间，如Schwartz缓增广义函数空间、Hida白噪声广义泛函空间等，也是有限维欧氏空间的自然推广；
2°其核空间结构使我们可以灵活地使用拓扑张量积、Bochner-Minlos定理、Schwartz核定理、It(?)正则化定理等，克服构造中的一系列困难。
以下是本文的几个主要结论：
结论一令E (?)H(?) E~*为由(H，A)生成实Gel'fand三元组。对于给定α∈E~*，Q∈(?)+和E~*上的测度v，其中v的支撑为E_(-p)且满足则存在q＞0和E~*上的无穷可分分布μ，使得我们称(α，Q，v)为μ的生成三元组，v为L(?)vy测度。
这样，我们得到了E~*上最一般的无穷可分分布特征泛函的表达式，特别地，我们构造了E~*上的稳定分布。然后，我们定义了E~*-值的L(?)vy过程，并给出其L(?)vy-It(?)分解。利用再生核Hilbert空间技巧，我们构造了一类特殊的算子值过程关于平方可积的L(?)vy过程的随机积分。
结论二基于E~*上无穷可分分布特征泛函的表达式，我们定义了E~*-值的L(?)vy白噪声。设X={X_t，t∈R}为(α，Q，v)生成的E~*-值L(?)vy过程，其中α∈E~*，Q∈(?)~+，L(?)vy测度v满足∫_(|x|_(-p)＞1)|x|_(-p)dv(x)＜∞，X_1的特征指数ψ由结论一给出。则对(?)f∈L~1(R)∩L~2(R)，为E~*-值R．V．使得则{(?)(f)，f∈S(R)}为E~*-值缓增白噪声，我们称其为E~*-值的L(?)vy白噪声。
结论三由结论二以及Riemann-Liouville分数积分算子I~β_的连续性，我们构造了一个新的E~*-值(缓增)广义过程作为E~*-值的L(?)vy白噪声的泛函，称为E~*-值的分数L(?)vy噪声。在结论二的条件下，定义则{(?)~β(f)，f∈S(R)}为E~*-值(缓增)广义过程(我们称为E~*-值β-分数L(?)vy噪声)。而且，特别地，对示性函数f=1_([0，t])，只要下面积分存在，我们就可以定义如下的E~*-值β-分数L(?)vy过程{X_t~β，t(?)0}：
结论四特别地，当E~*-值的L(?)vy过程X平方可积时，作为缓增的L(?)vy白噪声的泛函，我们研究了E~*-值的分数L(?)vy过程{X_t~β：=(?)~β(1_([0，t]))，t(?)0}的样本轨道性质和分布性质，证明了分数L(?)vy过程具有平稳增量性，长相依性质，对一类特殊的算子值过程讨论关于分数L(?)vy过程的随机积分，给出分数L(?)vy过程的新息表示。并将E~*-值分数L(?)vy过程作了推广，定义了组合分数L(?)vy过程和多分数L(?)vy过程。
结论五上述方法和结论可特别方便地推广到多维时间参数情形，在E~*上无穷可分分布的一阶矩存在的条件下，我们构造了E~*-值L(?)vy随机场。并利用Riesz位势，Riesz多重位势以及多变量型的Riemann-Liouville分数积分算子，构造了Gel，fand三元组上各向同性分数L(?)vy随机场，各向异性分数L(?)vy随机场以及多参数分数L(?)vy过程，并分别研究了它们的性质(Euclid不变性，长相依性，自相似性，平稳增量性等)。
By virtue of its self-similarity and long-range dependence,fractional Brownian mo-tion suits to model driving noises in different applications such as mathematical finance,economics,and network traffic analysis,and has evoked wide interests.Fractional Brown-ian motion can be represented as stochastic integral of a kernel function of Volterra typedetermined by the Riemann-Liouville fractional integral and derivative operator with re-spect to Brownian motion.By stochastic integration of this kernel function with respectto a general L(?)vy process,one can define fractional L(?)vy process which greatly widenthe fields of applications of fractional processes.
On the other hand,for the study on stochastic differential equations driven byfractional noises in infinitely dimensional vector spaces,one need to constuct fractionalprocesses in these spaces.So far as we know,the current study only limited to stochasticevolution equations driven by fractional Brownian motions in Hilbert spaces.However,the fractional Brownian motion B~H in a Hilbert space V exists if and only if its covarianceoperator Q is nuclear.Otherwise,it will take values in a larger Hilbert space V_1 suchthat V continuously imbedded into V_1 and the imbedding is a Hilbert-Schmidt operator.So,it is more natural to consider the fractional processes taking values in the dual spaceof a countably Hilbertian nuclear space.The most suitable framework should be theGel'fand triple.
Let H be a real separable Hilbert space with norm |·|_0 and inner product〈·,·〉.LetA be a positive self-adjoint operator in H such that A~(-α) is nuclear for someα＞0.Foreach r∈R,define |·|_r:=|A~r·|_0 and let E_r be the completion of the domain of A~rwith respect to |·|_r.Then E_r is a real Hilbert space,E_r and E_(-r) can be viewed as eachother's duals.Let E be the projective limit of {E_r}_r≥0 and E~* be the inductive limit of{E_(-r)}_r≥0.Then E is a Fr(?)chet nuclear space with E~* as its dual.We denote by〈·,·〉the canonical bilinear form on E~*×E,(?)~+≡(?)~+(E,E~*) the space of positive continuouslinear maps from E to E~*,then E(?)H(?)E~* is referred to as a Gel'fand triple generatedby (H,A).All the work of this dissertation is under this framework since
1°It includs many usual infinitely dimensional spaces such as Schwartz space of tempered distributions,Hida space of generalized white noise functionals,and is themost natural extension of finite dimensional Euclidean spaces;
2°Under this framework,we can freely use topological tensor product,Bochner-Minlos theorem,Schwartz kernel theorem,It(?) regularization theorem etc.to overcomemany difficulty in the construction.
The main conclusions of this dissertation are as following:
Conclusion 1 Let E(?)H(?)E~* be a real Gel'fand triple generated by (H,A) asabove.Given a∈E~*,Q∈(?)~+ and a Borel measure v on E~* supported in E_(-p) for somep＞0 satisfyingThen there exists q＞0 and an infinitely divisible distributionμon E~* such thatWe call (a,Q,v) the generating triple ofμand v the L(?)vy measure.
So,we obtain a general form of the charateristic functional of infinitely divisibledistributions on E~*,as a special case,we construct stable distributions on E~*.Thenwe define the E~*-valued L(?)vy process and give its L(?)vy-It(?) decomposition.Using areproducing kernel Hilbert space technique,we define stochastic integrals for a class ofoperator-valued processes with respect to L(?)vy processes.
Conclusion 2 By virtue of the charateristic functional of the infinitely divisibledistribution on E~*,we introduce E~*-valued L(?)vy white noises.Let X={X_t,t∈R} bean E~*-valued L(?)vy process generated by triple (a,Q,v) with a∈E~*,Q∈(?)~+ and L(?)vymeasure v satisfying f_(|x|-p＞1)|x|_(-p)dv(x)＜∞whose characteristic exponent is given byConclusion 1.Then,for any f∈L~1(R)∩L~2(R),is a well-defined E~*-valued R.V.such that Therefore,{(?)(f),f∈S(R)} is a tempered E~*-valued white noise.We call it L(?)vy whitenoise on Gel'fand triple E(?)H(?)E~*.
Conclusion 3 By Conclusion 2 and the continuity of the Riemann-Liouville frac-tional integration operator I_-~β,we construct a new E~*-valued (tempered) generalizedprocess as functional of E~*-valued L(?)vy white noise which is referred to as E~*-valuedfractional L(?)vy noise.Under the conditions of Conclusion 2,defineThen {(?)~β(f),f∈S(R)} is an E~*-valued (tempered) generalized process (referred to asE~*-valuedβ-fractional L(?)vy noise).Moreover,For f=1_([0,t]),we obtain the E~*-valuedβ-fractional L(?)vy process:whenever the right hand side exists.
Conclusion 4 When the underlying L(?)vy process is square integrable,we constructthe E~*-valuedβ-fractional L(?)vy process {X_t~β:= (?)~β(1_([0,t])),t≥0} and investigate itsdistribution and sample properties by considering it as functional of the E~*-valued L(?)vywhite noise.We prove that fractional L(?)vy process has stationary increments,long-range dependence,and give its innovational representation.In addition,we extend theconstruction to mixed fractional L(?)vy process and multi-fractional L(?)vy process.
Conclusion 5 The above ideas and conclusions could be easily generalized to thecase of higher dimensions of the time parameter.Under the condition that the firstmoment of the infinitely divisible distribution on E~* exists,replacing the operator I_-~βby Riesz potential,Riesz poly-potential and multi-variate fractional integration opera-tor respectively,we constuct the E~*-valued L(?)vy random field,the isotropic fractionalL(?)vy random fields and anisotropic fractional L(?)vy random fields,and investigate theirdistribution and sample properties such as Euclidean invariance,long-range dependence,self-similarity.
引文
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[29]Huang Z Y,Li C J.On fractional stable processes and sheets: white noise approach.J.Math.Anal.Appl.,2007,325: 624-635.
[30]李楚进．分式稳定过程及场的白噪声分析：[博士学位论文]．华中科技大学：图书馆，2005．
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[35]Maslowski B,Nualart D.Evolution equations diven by a fractional Brownian motion.J.Funct.Anal.,2003,202: 277-305.
[36]Tindel S,Tudor C A,Viens F.Stochastic evolution equations with fractional Brownian motion.Probab.Th.Rel.Fields,2003,127: 186-204.
[37]Duncan T E,Jakubowski J,Pasik-Duncan B.Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise.Stoch.Proc.Appl.,2005,115: 1357-1383.
[38]Wang C S,Qu M S,Chen J S.A white noise approach to infinitely divisible distributions on Gel'fand triple.J.Math.Anal.Appl.,2006,315: 425-435.
[39]黄志远，严加安．无穷维随机分析引论．科学出版社，1997．
[40]Treves F.Topological Vector Spaces,Distributions and Kernels.Academic Press,1967.
[41]Obata N.White Noise Analysis and Fock Space.Lect.Notes in Math.1557,Berlin: Springer-Verlag,1994.
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[43]Bertoin J.L(?)vy Processes.Cambridge: Cambridge University Press,1996.
[44]Sato K I.L(?)vy Processes and Infinite Divisibility.Cambridge: Cambridge University Press,1999.
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[52]Pipiras V,Taqqu M S.Deconvolution of fractional Brownian motion.J.Time Serie.Analy.,2002,23: 487-501.
[53]Jost C.Transformation formulas for fractional Brownian motion.Stoc.Proc.Appl.,2006,116: 1341-1357.
[54]Jost C.On the connection between Molchan-Golosov and Mandelbrot-Van Ness representation of fractional Brownian motion.J.Interal Equa.Appl.,2008,20:93-119
[55]Campbell J Y,Shiller R J.Stock Prices,Earnings and Expected Dividends.J.Finance,1998,43: 661-676.
[56]Daniel Kent,Hirshleifer D,Subrahmanyam A.Investor psychology and security market under-and overreactions.J.Finance,1998,53: 1839-1885.
[57]Wang X T,Wu M.Whitening filter and innovational representation of fractional Brownian motion.Chaos,Solitons and Fractals,2007,doi.: 10.1016/j.chaos.2007.07.092
[58]Cheridito P.Mixed fractional Brownian motion.Bernoulli,2001,7: 913-934.
[59]Cheridito P.Regularizing fractional Brownian motion with a view towards stock pricing.PhD theis,Zuich,2001.
[60]Filatova D.Mixed fractional Brownian motion: some related questions for computer network traffic modelling.International Conference on System and Electronic Systems,Krak(?)w,2008.
[61]Ei-Nouty C.The fractional mixed fractional Brownian motion.Stat.Prob.Letter,2003,65: 111-120.
[62]Marquardt T.Multivariate fractional integrated CARMA processes.To appear in Multivariate Anal.,2006.
[63]Peltier R F,L(?)vy-V(?)hel J.Multifractional Brown motion: definition and preliminary results,rapport de recharche de l'INRIA,1995,No.2645.
[64]Benassi A,Jaffard S,Roux D.Gaussian process and pseudodifferential elliptic operators.Rev.Mat.Iberoamericana,1997,13: 19-89.
[65]Ayache A,L(?)vy-V(?)hel J.Generalized multifractional Brown motion: definition and preliminary results,In: Dekking M,L(?)vy-V(?)hel J,Lutton E,et al.(eds),Fractals: Theory and Applications in Engineering,London: Springer,1999.
[66]Cohen S.From self-similarity to local self-similarity: the estimation problem,in: Dekking M,L(?)vy-V(?)hel J,Lutton E,et al.Fractals: Theory and Application in Engineering,Springer,1999.
[67]Cohen S,Marty R.Invariance principle multifractional Gaussian processes and longrange dependence.eprint arXiv:math/0610551,2006.
[68]Ayache A,L(?)vy-V(?)hel J.The generalized multifractional Brown motion.Stat.Infere.Stoch.proc.,2000,3: 7-18.
[69]Benassi A,Cohen S,Istas J.Identifying the multifractional function of a Gaussian process.Stat.Prob.Letter,1998,39: 337-345.
[70]Ayache A,Cohen S,L(?)vy-V(?)hel J.The covariance structure of multifractional Brownian motion,with application to long range dependence,acte de la conf(?)rence ICASSP,Istambul,2000.
[71]Ayache A.The generalized multifractional field: a nice tool for the study of the generalized multifractional Brown motion.J.Fourier Analy.Appli.,2002,8: 581-601.
[72]Herbin E.From N-parameter fractional Brown motion to N-parameter multifractional Brown motion.Rock.Moun.J.Math.,2006,36(4): 1249-1284.
[73]Lacaux C.Real harmonizable multifractional L(?)vy motion.Ann.Inst.Poicar.Prob. Stat.,2004,40(3): 259-277.
[74]Stein E M.Singular Integrals and Differentiability Properties of Functions.Princeton: Princeton University Press,1970.
[75]Huang Z Y,Li C J.Anisotropic fractional random fields as white noise functionals.Acta Math.Appl.Sinica,2005,21(4): 655-660.
[76]Huang Z Y.Fractional generalized L(?)vy random fields as white noise functionals.Front.Math.of China,2007,2(2): 211-226.
[77]Nunno G D,φksendal B,Proske F.White noise analysis for L(?)vy processes.J.Funct.Anal.,2004,206: 109-148.
[78]Lee Y J,Shih H H.L(?)vy white noise measures on infinite-dimensional spaces: Existence and characterization of the measurable support.J.Funct.Anal.,2006,237: 617-633.
[79]Tornφe C W,Overgaard R V,Agersφ H,et al.Stochastic differential equations in NONMEM: Implementation,application,and comparision with ordinary differential equations.J.Pharm.Res.,2005,22: 1247-1258.
[80]Kristensen N R,Madsen H,Ingwersen S H.Using stochastic differential equations for PK\PD model development.J.Pharmacokinet.Pharmacodyn.,2005,32: 109-141.
[81]Bass R F,Pyke R.The existence of set-indexed L(?)vy processes.Z.Wahrscheinlichkeits Theorie Verw,1984,66: 157-172.
[82]Balan R M.Set-indexed processes with independent increments.Stat.and Proba.Lett.,2002,59: 415-424.
[83]Herbin E,Merzbach E.A set-indexed fractional Brown motion.J.of Theor.Proba.,2006,19(2): 337-364.
[84]Herbin E,Merzbach E.A characterization of the set-indexed fractional Brown motion by increasing paths.eprint arXiv:math/0607575,2006.
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