几种自相似高斯随机系统的分析及相关问题
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摘要
本学位论文旨在研究几种自相似高斯随机系统的分析及相关问题,全文共分七章。
第一章主要介绍分数布朗运动,次分数布朗运动,双分数布朗运动的基本概念及相关性质。给出了问题产生的背景、研究现状以及本学位论文研究所需要的预备知识。
第二章利用Malliavin分析的技巧发展了次分数布朗运动SH,H>1/2的随机积分。第一节介绍了次分数布朗运动的Malliavin分析,第二节给出了对称积分存在的充分条件,建立了对称积分与散度积分之间的关系(定理2.1),得到了散度积分的Lp最大值不等式(定理2.2)和1/H变差(定理2.3),推导了积分过程的It6公式(定理2.4),最后得到了一维次分数布朗运动关于C2函数的It6公式(定理2.5),然后将研究结果推广到多维次分数布朗运动情形(定理2.6,定理2.7)。
第三章主要研究指数H∈(1/2,1)的次分数布朗运动SH的随机分析,得到在不同条件下的It6公式。第一节得到了含有赋权局部时LH(x,t)的Tanaka公式(定理3.1),第二节证明了如果函数f是有界p(1≤p<(2H)/(1-H))变差的,那么积分∫R f(x) LH(dx,t)有意义(命题3.4)。作为一个应用,证明了对任意绝对连续函数F(x)=F(0)+∫0x f(y)dy,其中函数f是有界p(1≤p<(2H)/(1-H))变差的且是左连续的,有Bouleau-Yor公式(定理3.3)成立,第三节研究了f(SH)和SH的赋权二次协变差[f(SH),SH](W):其中极限依概率一致成立,x(?)f(x)是一个确定的函数。证明了如果函数f是有界p(1≤p<(2-H)/(1-H))变差的,那么赋权二次协变差[f(SH),SH](W)存在且有Bouleau-Yor等式(定理3.4)
第四章主要研究次分数布朗运动和双分数布朗运动的相遇局部时和相交局部时,得到了它们的存在性,光滑性,正则性等。第一节介绍了随机变量的光滑性,第二节证明了次分数布朗运动满足局部非确定性(定理4.1),第三节研究了指标分别为H1,H2的两个相互独立的次分数布朗运动的相遇局部时的存在性(定理4.2),利用初等方法给出了相遇局部时在Meyer-Watanabe意义下是光滑的充要条件(定理4.4),第四节得到了Rd,d≥2上指标均为H∈(0,1)的两个独立次分数布朗运动的相交局部时在L2中存在的充要条件(定理4.5)以及它在Meyer-Watanabe意义下是光滑的充要条件(定理4.6),最后研究了相交局部时的正则性(定理4.7),第五节主要利用初等方法得到了两个相互独立且指标不同的双分数布朗运动的相遇局部时在Meyer-Watanabe意义下是光滑的充要条件(定理4.8),改进了Jiang-Wang[47](也可参见Yan et al[48])中的相应结果,同时也研究了指标相同的两个相互独立双分数布朗运动的相交局部时在Meyer-Watanabe意义下是光滑的充要条件(定理4.9)。
第五章主要研究次分数布朗运动的一些相关过程。第一节研究了随机过程Xt：=∑id=1∫0t(Ss1)/(Rs)dSsi,d≥1的一些性质(自相似性(命题5.1和命题5.7),短相依性(定理5.1和定理5.3)等),其中H≥1/2,Rt=(?)是次分数Bessel过程,给出了随机过程Zt的Wiener混沌展开(定理5.2)以及随机过程Rt的积分表现(命题5.6)等,第二节研究了次分数布朗运动驱动的积分泛函Xt(j):=Aj(t,StH)-∫0t Lj(s,SsH)dSsH的p-变差,其中H≥1/2,)(?)j(t,x),j=1,2分别是SH在x点的局部时与赋权局部时,找到了一个仅依赖于H的常数pH使得当p>pH时,Xt(j)的p-变差等于零(定理5.7)。
第六章首先构造了一簇连续随机过程(In∈(f)t)∈>0：证明了(In∈(f)t)∈>0依有限维分布收敛到关于次分数布朗运动的多重Wiener-Ito积分过程(InH,e(f1[0,t](?)n))t∈[0,1](定理6.1,定理6.2,定理6.3),其中被积函数f∈|H|(?)n,η∈(t)=∫0tθ∈(x)dx是Donsker或Stroock逼近。其次研究了与次分数布朗运动相关的两个极限定理(定理6.5和定理6.6)等.
第七章利用双分数布朗运动的Malliavin分析技巧研究了当n→∞时序列的渐近行为,其中BH1,K1和BH2,K2是两个独立的双分数布朗运动,K是一个核函数,带宽参数α满足关于H1,K1和H2,K2的一些假设,证明了它的极限分布是包含双分数布朗运动BH1,K1局部时的一个混合正态分布(定理7.3),研究了向量(Sn,(Gt)t≥0)的收敛性(定理7.6),其中(Gt)t≥0是与BH1,K1独立的随机过程且满足一些附加条件。
This dissertation aims to study the stochastic calculus for some self-similar Gaussian sys-tems and related topics. It consists of seven chapters.
In Chapter 1, we introduce some preliminary concepts and necessary properties about fractional Brownian motion, subfractional Brownian motion (sub-fBm in short) and bifractional Brownian motion(bi-fBm in short).
In Chapter 2, we develop a stochastic calculus for the sub-fBm SH,H>1/2using the tech-niques of the Malliavin calculus. In Section 2.1 we present Malliavin calculus for sub-fBm. In Section 2.2, we gives sufficient conditions for the existence of the symmetric integral and establish the relationship between the symmetric integral and the divergence integral(Theorem 2.1), we establish estimates in LP maximal inequalities(Theorem 2.2) and 1/H variation (Theo-rem 2.3) for the stochastic integral and derive an Ito s formula for integral processes(Theorem 2.4), we also derive an Ito formula for sub-fBm(Theorem 2.5) and extend Ito's formula to the multidimensional case (Theorem 2.6, Theorem 2.7).
In Chapter 3, we consider stochastic calculus connected with sub-fBm SH with H∈(1/2,1) and narrow the focus to obtain various versions of Ito's formula. In Section 3.1 the Tanaka formula (Theorem 3.1) is obtained and it involves the so-called weighted local time LH(x, t). In Section 3.2, we show that the integral∫R f(x)LH(dx, t) is well-defined provided f∈Wp with 1≤p<(2H)/(1-H)(Proposition 3.4). As an application we show that Bouleau-Yor's formula holds for all absolutely continuous function F(x)=F(0)+∫0x f(y)dy, where the derivative f∈Wp be a left continuous function with 1≤p<(2H)/(1-H). In Section 3.3, we study the weighted quadratic covariation [f(SH), SH](W) of f(SH) and SH defined by for t≥0, where the limit is uniform in probability and x(?) f(x) is a deterministic function. We show that the weighted quadratic covariation [f(SH),SH](W) exists and if f∈WP with 1≤p<(2H)/(1-H) (Theorem 3.4).
In Chapter 4, we consider the collision local time and intersection local time of sub-fBm and bi-fBm and obtain the existence, smoothness and regularity of local time. In Section 4.1, we introduce the definition of smooth of random variation, In Section 4.2, we obtain the local nondeterminism of sub-fBm(Theorem 4.1). In Section 4.3, we study the collision local time of two independent sub-fBms SHi, i=1,2 with respective indices Hi∈(0,1) and obtain the existence(Theorem 4.2) and by an elementary method we show that it is smooth in the sense of Meyer and Watanabe if and only if min{H1, H2}<1/3 (Theorem 4.4). In Section 4.3, we study the intersection local time of two independent d-dimensional sub-fBms SH and SH with indices H∈(0,1), and we show that the intersection local time exists in L2 if and only if Hd<2 (Theorem 4.5) and give the necessary and sufficient conditions for which it is smooth in the sense of the Meyer-Watanabe(Theorem 4.6). As a related problem, we give also the regularity of the intersection local time process(Theorem 4.7). In Section 4.4, we use an elementary method to prove the necessary and sufficient conditions of the smooth of the collision local time process for two independent bi-fBms BHi,Ki,i=1,2 with respective indices Hi∈(0,1), Ki∈(0,1] (Theorem4.8), the results extend and improve the corresponding theorems in Jiang-Wang [47](also Yan et al [48]). At last, we study the intersection local time of two independent bi-fBms BH'K and BH'K with same indices H∈(0,1), K∈(0,1], we show that it is smooth in the sense of the Meyer-Watanabe if and only if HK<2/3 (Theorem 4.9).
In Chapter 5, we consider some related process of sub-fBm. In Section 5.1, we study some properties (Proposition 5.1, Proposition 5.7, Theorem 5.1, Theorem 5.3) of the process X of the form Xt:=(?)≥1 where H>1/2, Rt= (?) is the sub-fractional Bessel process, we obtain the chaos expansion of Zt(Theorem 5.2) and give an integral representation for sub-fractional Bessel processes(Proposition 5.6). In Section 5.2, we show that there exists a constant pH such that p-variation of the process Aj(t, StH)-∫0t Lj(s, SsH)dSsH (j=1,2) equals to 0 if p>pH, where Lj,j=1,2, are the local time and weighted local time of SH, respectively(Theorem 5.7).
In Chapter 6, we prove the family (In∈(f))∈>0 defined by converges in law to the multiple Wiener-Ito integrals (InH,e(f1[0,t](?)n))t∈[0,1] with respect to the sub-fBm, for the integrand f∈|H|(?)n, whereη∈(t)=∫0tθ∈(x)dx are Donsker and Stroock approximations (Theorem 6.1, Theorem 6.2, Theorem 6.3). The limit theorems associated to the sub-fBm SH are also discussed (Theorem 6.5, Theorem 6.6).
In chapter 7, we use the techniques of the Malliavin calculus with respect to the bi-fBm to study the asymptotic behavior as n→∞of the sequence where BH1,K1 and BH2,K2 are two independent bi-fBms, K is a kernel function and the band-width parameterαsatisfies certain hypotheses in terms of H1, K1 and H2, K2. We prove its lim-iting distribution is a mixed normal law involving the local time of the bi-fBm BH1,K1 (Theorem 7.3), we also study the convergence of the vector (Sn, (Gt)t≥0), where (Gt)t≥0 is a stochastic process independent from BH1,K1 and satisfies some additional conditions(Theorem 7.6).
第一章主要介绍分数布朗运动,次分数布朗运动,双分数布朗运动的基本概念及相关性质。给出了问题产生的背景、研究现状以及本学位论文研究所需要的预备知识。
第二章利用Malliavin分析的技巧发展了次分数布朗运动SH,H>1/2的随机积分。第一节介绍了次分数布朗运动的Malliavin分析,第二节给出了对称积分存在的充分条件,建立了对称积分与散度积分之间的关系(定理2.1),得到了散度积分的Lp最大值不等式(定理2.2)和1/H变差(定理2.3),推导了积分过程的It6公式(定理2.4),最后得到了一维次分数布朗运动关于C2函数的It6公式(定理2.5),然后将研究结果推广到多维次分数布朗运动情形(定理2.6,定理2.7)。
第三章主要研究指数H∈(1/2,1)的次分数布朗运动SH的随机分析,得到在不同条件下的It6公式。第一节得到了含有赋权局部时LH(x,t)的Tanaka公式(定理3.1),第二节证明了如果函数f是有界p(1≤p<(2H)/(1-H))变差的,那么积分∫R f(x) LH(dx,t)有意义(命题3.4)。作为一个应用,证明了对任意绝对连续函数F(x)=F(0)+∫0x f(y)dy,其中函数f是有界p(1≤p<(2H)/(1-H))变差的且是左连续的,有Bouleau-Yor公式(定理3.3)成立,第三节研究了f(SH)和SH的赋权二次协变差[f(SH),SH](W):其中极限依概率一致成立,x(?)f(x)是一个确定的函数。证明了如果函数f是有界p(1≤p<(2-H)/(1-H))变差的,那么赋权二次协变差[f(SH),SH](W)存在且有Bouleau-Yor等式(定理3.4)
第四章主要研究次分数布朗运动和双分数布朗运动的相遇局部时和相交局部时,得到了它们的存在性,光滑性,正则性等。第一节介绍了随机变量的光滑性,第二节证明了次分数布朗运动满足局部非确定性(定理4.1),第三节研究了指标分别为H1,H2的两个相互独立的次分数布朗运动的相遇局部时的存在性(定理4.2),利用初等方法给出了相遇局部时在Meyer-Watanabe意义下是光滑的充要条件(定理4.4),第四节得到了Rd,d≥2上指标均为H∈(0,1)的两个独立次分数布朗运动的相交局部时在L2中存在的充要条件(定理4.5)以及它在Meyer-Watanabe意义下是光滑的充要条件(定理4.6),最后研究了相交局部时的正则性(定理4.7),第五节主要利用初等方法得到了两个相互独立且指标不同的双分数布朗运动的相遇局部时在Meyer-Watanabe意义下是光滑的充要条件(定理4.8),改进了Jiang-Wang[47](也可参见Yan et al[48])中的相应结果,同时也研究了指标相同的两个相互独立双分数布朗运动的相交局部时在Meyer-Watanabe意义下是光滑的充要条件(定理4.9)。
第五章主要研究次分数布朗运动的一些相关过程。第一节研究了随机过程Xt：=∑id=1∫0t(Ss1)/(Rs)dSsi,d≥1的一些性质(自相似性(命题5.1和命题5.7),短相依性(定理5.1和定理5.3)等),其中H≥1/2,Rt=(?)是次分数Bessel过程,给出了随机过程Zt的Wiener混沌展开(定理5.2)以及随机过程Rt的积分表现(命题5.6)等,第二节研究了次分数布朗运动驱动的积分泛函Xt(j):=Aj(t,StH)-∫0t Lj(s,SsH)dSsH的p-变差,其中H≥1/2,)(?)j(t,x),j=1,2分别是SH在x点的局部时与赋权局部时,找到了一个仅依赖于H的常数pH使得当p>pH时,Xt(j)的p-变差等于零(定理5.7)。
第六章首先构造了一簇连续随机过程(In∈(f)t)∈>0：证明了(In∈(f)t)∈>0依有限维分布收敛到关于次分数布朗运动的多重Wiener-Ito积分过程(InH,e(f1[0,t](?)n))t∈[0,1](定理6.1,定理6.2,定理6.3),其中被积函数f∈|H|(?)n,η∈(t)=∫0tθ∈(x)dx是Donsker或Stroock逼近。其次研究了与次分数布朗运动相关的两个极限定理(定理6.5和定理6.6)等.
第七章利用双分数布朗运动的Malliavin分析技巧研究了当n→∞时序列的渐近行为,其中BH1,K1和BH2,K2是两个独立的双分数布朗运动,K是一个核函数,带宽参数α满足关于H1,K1和H2,K2的一些假设,证明了它的极限分布是包含双分数布朗运动BH1,K1局部时的一个混合正态分布(定理7.3),研究了向量(Sn,(Gt)t≥0)的收敛性(定理7.6),其中(Gt)t≥0是与BH1,K1独立的随机过程且满足一些附加条件。
This dissertation aims to study the stochastic calculus for some self-similar Gaussian sys-tems and related topics. It consists of seven chapters.
In Chapter 1, we introduce some preliminary concepts and necessary properties about fractional Brownian motion, subfractional Brownian motion (sub-fBm in short) and bifractional Brownian motion(bi-fBm in short).
In Chapter 2, we develop a stochastic calculus for the sub-fBm SH,H>1/2using the tech-niques of the Malliavin calculus. In Section 2.1 we present Malliavin calculus for sub-fBm. In Section 2.2, we gives sufficient conditions for the existence of the symmetric integral and establish the relationship between the symmetric integral and the divergence integral(Theorem 2.1), we establish estimates in LP maximal inequalities(Theorem 2.2) and 1/H variation (Theo-rem 2.3) for the stochastic integral and derive an Ito s formula for integral processes(Theorem 2.4), we also derive an Ito formula for sub-fBm(Theorem 2.5) and extend Ito's formula to the multidimensional case (Theorem 2.6, Theorem 2.7).
In Chapter 3, we consider stochastic calculus connected with sub-fBm SH with H∈(1/2,1) and narrow the focus to obtain various versions of Ito's formula. In Section 3.1 the Tanaka formula (Theorem 3.1) is obtained and it involves the so-called weighted local time LH(x, t). In Section 3.2, we show that the integral∫R f(x)LH(dx, t) is well-defined provided f∈Wp with 1≤p<(2H)/(1-H)(Proposition 3.4). As an application we show that Bouleau-Yor's formula holds for all absolutely continuous function F(x)=F(0)+∫0x f(y)dy, where the derivative f∈Wp be a left continuous function with 1≤p<(2H)/(1-H). In Section 3.3, we study the weighted quadratic covariation [f(SH), SH](W) of f(SH) and SH defined by for t≥0, where the limit is uniform in probability and x(?) f(x) is a deterministic function. We show that the weighted quadratic covariation [f(SH),SH](W) exists and if f∈WP with 1≤p<(2H)/(1-H) (Theorem 3.4).
In Chapter 4, we consider the collision local time and intersection local time of sub-fBm and bi-fBm and obtain the existence, smoothness and regularity of local time. In Section 4.1, we introduce the definition of smooth of random variation, In Section 4.2, we obtain the local nondeterminism of sub-fBm(Theorem 4.1). In Section 4.3, we study the collision local time of two independent sub-fBms SHi, i=1,2 with respective indices Hi∈(0,1) and obtain the existence(Theorem 4.2) and by an elementary method we show that it is smooth in the sense of Meyer and Watanabe if and only if min{H1, H2}<1/3 (Theorem 4.4). In Section 4.3, we study the intersection local time of two independent d-dimensional sub-fBms SH and SH with indices H∈(0,1), and we show that the intersection local time exists in L2 if and only if Hd<2 (Theorem 4.5) and give the necessary and sufficient conditions for which it is smooth in the sense of the Meyer-Watanabe(Theorem 4.6). As a related problem, we give also the regularity of the intersection local time process(Theorem 4.7). In Section 4.4, we use an elementary method to prove the necessary and sufficient conditions of the smooth of the collision local time process for two independent bi-fBms BHi,Ki,i=1,2 with respective indices Hi∈(0,1), Ki∈(0,1] (Theorem4.8), the results extend and improve the corresponding theorems in Jiang-Wang [47](also Yan et al [48]). At last, we study the intersection local time of two independent bi-fBms BH'K and BH'K with same indices H∈(0,1), K∈(0,1], we show that it is smooth in the sense of the Meyer-Watanabe if and only if HK<2/3 (Theorem 4.9).
In Chapter 5, we consider some related process of sub-fBm. In Section 5.1, we study some properties (Proposition 5.1, Proposition 5.7, Theorem 5.1, Theorem 5.3) of the process X of the form Xt:=(?)≥1 where H>1/2, Rt= (?) is the sub-fractional Bessel process, we obtain the chaos expansion of Zt(Theorem 5.2) and give an integral representation for sub-fractional Bessel processes(Proposition 5.6). In Section 5.2, we show that there exists a constant pH such that p-variation of the process Aj(t, StH)-∫0t Lj(s, SsH)dSsH (j=1,2) equals to 0 if p>pH, where Lj,j=1,2, are the local time and weighted local time of SH, respectively(Theorem 5.7).
In Chapter 6, we prove the family (In∈(f))∈>0 defined by converges in law to the multiple Wiener-Ito integrals (InH,e(f1[0,t](?)n))t∈[0,1] with respect to the sub-fBm, for the integrand f∈|H|(?)n, whereη∈(t)=∫0tθ∈(x)dx are Donsker and Stroock approximations (Theorem 6.1, Theorem 6.2, Theorem 6.3). The limit theorems associated to the sub-fBm SH are also discussed (Theorem 6.5, Theorem 6.6).
In chapter 7, we use the techniques of the Malliavin calculus with respect to the bi-fBm to study the asymptotic behavior as n→∞of the sequence where BH1,K1 and BH2,K2 are two independent bi-fBms, K is a kernel function and the band-width parameterαsatisfies certain hypotheses in terms of H1, K1 and H2, K2. We prove its lim-iting distribution is a mixed normal law involving the local time of the bi-fBm BH1,K1 (Theorem 7.3), we also study the convergence of the vector (Sn, (Gt)t≥0), where (Gt)t≥0 is a stochastic process independent from BH1,K1 and satisfies some additional conditions(Theorem 7.6).
引文
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