Statistical Inference For The Ornstein-Uhlenbeck Process Based On Malliavin Calculus

Ornstein-Uhlenbeck(O-U)type processes,as a class of important diffusion processes,have a wide range of applications in the research field of physics and finance.In physics,they are often applied to describe the evolution process of dynamic systems disturbed by stochastic noise.For example,as the solution of Langevin equation,O-U process can simulate the speed of particles in the Coulonb gas model.In finance,O-U type processes are usually applied to study the fluctuations of interest rates and exchange rates.For instance,we can use Hull-While model to describe the bonds pricing and evaluate their risks.However,there usually exist some unknown parameters in the above mentioned processes.Therefore,for practical applications,it is necessary to study the precise asymptotic properties of the estimators related to the unknown parameters.Based on the review and summary of the existed works,for the O-U process without tears and fractional O-U process with periodic mean,we study the asymptotic properties of the estimators in their drift coefficients(deviation inequalities,moderate deviation principles and Cramér-type moderate deviations).The main methods of this thesis consist of Malliavin calculus,deviation inequalities and moderate deviation principles for multiple Wiener-It? integrals,and asymptotic analysis techniques.The main results of this thesis include the following two parts.1.For the trajectory fitting estimator(TEF)in the O-U process without tears,deviation inequalities,moderate deviation principles and Berry-Esseen bounds could be obtained.The main methods include the deviation inequality and moderate principle for multiple Wiener-It? integrals,and asymptotic analysis techniques.2.For the least squares estimator(LSE)in the fractional O-U process with period mean,we study its Cramér-type moderate deviations,and moderation deviation principles.Furthermore,it is interesting to note that we find a phase transition phenomena related to the Hurst parameter.In this part,besides methods of deviation inequalites and moderate deviaton principles for multiple Wiener-It? integrals,we also apply the theory of Malliavin calculus related to the fraction Brownian motion.