Local Time Of Fractional Brownian Motion And Stochastic Analysis For Related Processed

This dissertation aims to study fractional Brownian motion and some related self-similar stochastic processes, especially investigate the local time-space calculus,weighted local time and intersection local time of fractional Brownian motion. Through the analysis of their path-wise, distribution and so on, we get a lot of meaningful results, which also enrich contents of self-similar Gaussian process. Moreover, we discuss some related self-similar processes in-cluding sub-fractional Brownian motion, Rosenblatt process and iterated process. The paper divides into seven chapters.In Chapter 1, we first introduce some basic concepts and related properties about fractional Brownian motion. Then, we establish some interesting inequalities associated with Bernoulli's inequality, they play a auxiliary role in the following chapters.In Chapter 2, for Hurst index H∈(0,1/2), we study the generalized quadratic covariation [f(BH), BH](?) of fractional Brownian motion. We get the condition ensures it exist in L2 and obtain the generalized Ito's formula: Unlike when H∈(1/2 1), fractional Brownian motion has time reversal process, so our start point is to consider the decomposition then we construct a Banach space (?) of measurable functions and show that the generalized quadratic covariation exists in L2 if f∈(?). Next, we study the local time-space integral fR f(x)(?)H(dx, t) and obtain corresponding Bouleau-Yor type identity with the form provided f∈(?). This allows us to write the fractional Ito's formula for absolutely contin-uous functions with derivative belonging to (?).These are also can be extended to the time-dependent case. In Chapter 3, for Hurst index H∈(1/2,1), we investigate the weighted local time(?)H(t, x) of fractional Brownian motion. Based on fractional Clark-Ocone formula and a tool analogous to the asymptotic version of Knight's theorem, for the L2 modulus of continuity of increment process of (?) H(t, x) with respect to space x we establish a beautiful central limit theorem, that is, for each fixed t≥0, holds as h tends to zero, whereηis a N(0,1) random variable independent of Bt H.In Chapter 4, for Hurst index H∈(0,1), we study the intersection local time of two inde-pendent fractional Brownian motion defined on Rd, d≥2. We state the background knowledge of intersection local time and introduce the chaos expansion of L2 space in Section 4.1. In Sec-tion 4.2, using the interesting inequalities established in Section 1.2, we consider the sufficient and necessary condition of its existence in L2 through elementary methods. In last section, we show that it is smooth in the sense of Meyer-Watanabe if and only if H≤d+2/2.In Chapter 5, we study the stochastic analysis of another process associated with fractional Brownian motion-sub-fractional Brownian motion with index H∈(0,1/2) and obtain several generalized Ito's formula. Firstly, we present some preliminaries for sub-fBm and establish some technical estimates, it seems interesting that these inequalities arising from the method. Next, we construct a new Banach space (?), such that the generalized quadratic covariation [f(SH), SH] (W) exists in L2 for f∈(?), as an application we show that the Ito type formula (Follmer-Protter-Shiryayev's formula) holds, where F is an absolutely continuous function with the derivative F'=f∈(?). At last, we introduce the integral with respect to local time of the form We show that the integral exists in L2 and establish the Bouleau-Yor type identity and Tanaka formula:In Chapter 6, we consider another important Hermite processes-Rosenblatt process. Using a sequence square integrable martingale differences we define a new stochastic process Zn and prove an approximation theorem for Rosenblatt processes Z with H∈(1/2,1) as n tends to infty.In Chapter 7, we study iterated process{I(t)}t≥0 generated by multi-dimension fractional Brownian motion and multi-parameter fractional Brownian sheet. Through some basic heat equations and complex integral calculus, we find their probability density function can be the solution of a few partial differential equations.