Stochastic Calculus For Fractional Lévy Process And Its Related Processes

This dissertation mainly studies the stochastic calculus for fractional Lévy process(FLP),?-stable Lévy process and multifractional stable sheets.It consists of four chapters.In chapter 1,we introduce some preliminary conceptions of FLP,Ornstein Uhlenbeck(O-U)process of second kind,multifractional stable sheets and local times.We also offer the necessary properties of FLP and ?-stable Lévy process.In chapter 2,we consider the problem of parameter estimation for O-U process with small fractional Lévy noise,based on discrete observations at n regularly spaced time points ti = i/n,i = 1,...,n on [0,1].Least squares method is used to obtain an estimator of the drift parameter.The consistency and the asymptotic distribution of the estimator have been established.In chapter 3,we study the problem of parameter estimation for O-U process of the second kind driven by ?-stable Lévy process,based on continuous and discrete observations,respectively.Using the trajectory fitting method combined with the weighted least squares technique,we discuss the consistency and the asymptotic distributions of the estimators for general weights in both the ergodic and the non-ergodic cases.In chapter 4,let XH(u)(u)= {XH(u)(u),u ? RN+} be multifractional stable sheets with index functional H(u),where H(u)=(H1(u),...,HN(u))is a function with values in(0,1)N.Based on some assumptions of H(u),we obtain the existence of the local times of XH(u)(u)and establish its joint continuity and the H¨older regularity.