Stochastic Calculus For Some Self-similar Gaussian Systems And Related Topics

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, where??(t)=?0t??(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 parameter?satisfies 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).