Research On FBm-like Gaussian Processes And Their Driving Stochastic Differential Equations

In this thesis,we mainly study limit theorems for functionals of fBm-like Gaussian processes and parameter estimation of stochastic differential equations driven by these Gaussian processes.The full text is divided into four chapters.In Chapter 1,we present some basic concepts of fBm-like Gaussian processes,such as fractional Brownian motion,sub-fractional Brownian motion and bi-fractional Brownian motion,and recall the research techniques of local nondeterminism property,Wiener chaos expansion and Malliavin calculus.We also analyze the research status of limit theorems for functionals of fBm-like Gaussian processes and parameter estimation of stochastic differential equations driven by them.In Chapter 2,we study some limit theorems for functionals of two independent fBm-like Gaussian processes,under certain conditions.We mainly use the method of moments combined with Fourier analysis,chaining argument and a pairing technique,and give the effect of non-stationary increments on the limit distribution.In Chapter 3,we study the derivatives of self-intersection local time for multidimensional fractional Brownian motion,and give the sufficient condition for the existence in L2 sense and its Holder continuity conditions.Especially,based on the Wiener chaos decomposition we show a limit theorem for the critical case with H=2/3 which was conjectured in Jung?Markowsky(2014).We find that only the first order chaos plays an important role in the limit theorem,then we continue to study the limit distribution of arbitrary chaos.In Chapter 4,we discuss strong consistency for least squares estimator of the drift parameters of fractional Ornstein-Uhlenbeck process with periodic mean function for all the Hurst parameter range H ?(0,1).With different normalization factors,two central limit theorems are proved when H<1/2 and H>1/2,respectively.Moreover,we also discuss the asymptotic behavior for quadratic variation of stochastic integral driven by fBm-like Gaussian processes.As an application,we construct a strongly consistent estimator for the integrated volatility parameter in stochastic differential equations driven by these Gaussian processes.