Parameter Estimation Of Ornstein-Uhlenbeck Process And Its Property

Ornstein-Uhlenbeck process is a diffusion process as well as Markov process. It alsohas a wide range of applications in physics and financial mathematics. In this paper, wediscuss the parameter estimation of Ornstein-Uhlenbeck process and its properties,including characteristic function and the Iterated Logarithm law. Firstly, we estimate σ~2by Iterated Logarithm Law for wiener integral of square integrable function as well asGlen Baxter theorem; and we give the estimation of a by ruler method as well asapproximate maximum likelihood method. However, the time intervals between all thepairs of financial data we have are not small enough to0, therefore we apply the pathintegration method, and implement the estimation of a by means of MATLAB. Theestimation obtained by path integral gets close to the exact value, but the accuracy and theprogram speed need further development.The article is divided into ifve parts. The ifrst chapter is the introduction patr,including the research significance and the development of OU process. The secondchapter gives a mathematical model of Ornstein-Uhlenbeck process. Besides, we recallsome related basic knowledge of Stochastic Analysis. Chapter three discusses thecharacteristic function and iterated logarithm law of Ornstein-Uhlenbeck process. Chapterfour and chapter ifve are the main body of the paper. Chapter four gives the parameteralpha, sigma’s estimate expression by use of iterated logarithm law, Glen Baxter theorem,Euler method as well as maximum-likelihood method. Chapter ifve introduces the pathintegration, which means to obtain the transition probability density of the sampleinterpolation, MATLAB contributes to the implementation of path integration, whichhelps to obtain some good approximation.