Research On Parameter Estimation And Applications For Several Fractional Ornstein-uhlenbeck Processes

The fractional Ornstein-Uhlenbeck process(fractional O-U process in short), which is the solution of a one-dimensional homogeneous linear stochastic differential equation dXt=θXtdt+dWtH, where WtH is fractional Brownian motion for Hurst index H∈[1/2,1) Firstly, O-U process is driven by a Brownian motion, which is extensively used in physics, finance fields, such as we use it to describe the fluctuation of exchange rate and interest fluctuation. But financial empirical study show that:stock price has rush back and long-rang dependence, and Brownian motion has not these properties. That is to say, the differential equation models driven by Brownian motion can not character stock price perfectly. So promoting classical O-U processes to fractional O-U processes is necessary. Owing to fractional Brownian motion and sub-fractional Brownian motion are general Brownian motions, and have self-similarity, long-range dependence and so on. We use fractional O-U process model and sub-fractional O-U process model to simulate random phenomena, which are close to the actual. When the two models are used to describe random phenomena, it is very necessary to identify the unknown parameters in these models. Meanwhile, there are financial implications considering them as exchange rate and interest rate models.This paper is mainly divided into two parts. We use the spectral density method to estimate the drift parameter θ of fractional O-U process in the first section. Firstly, we discrete the simple linear stochastic differential equation dXt=θXtdt+dWtH, and the fractional Gaussian noise is approximated by the Gaussian process which is expressed by a spectral density, then we construct a maximum likelihood estimator (MLE) for the drift parameter θ,and discuss the properties of the estimation value:unbiased property, approximate normality and strong consistency. We take Ping An bank stock closing price as an example, comprising the fractional Brownian motion models estimated by stochastic representation method and spectral density method, and do the empirical analysis.In the second section we consider the following stochastic differential equation dXt=θXtdt+dStH, where StH is sub-fractional Brownian motion for Hurst index H ∈[1/2,1). We use the minimum contrast method to estimate the drift parameter θ of sub-fractional O-U process. Firstly taking derivative of Radon-Nikodym, and we obtain a contrast function using sub-fractional Ito equation, then we estimate the drift parameter θ, and discuss the strong consistency of the estimation value. We use Monte Carlo method to simulate and prove unbiased property and efficiency of the estimation value, and comprise with the MLE.The paper use spectral density method and minimum contrast method to get the drift parameter of fractional O-U process and sub-fractional O-U process, and not only provide the parameter estimation methods in the models, but also use the models by estimated to simulate random variation path of the future stock price, and calculate VaR to provide decision basis for financial market.