The Asymptotic Behavior And Correlation Analysis Of The Fractional Self-attracting Diffusion

In this article we mostly study the discrete version of the linear fractional self-attracting diffusion driven by fractional Brown Motion and the estimate of the parameter and its correlation analysis. So-called self-attracting diffusion is the following stochastic differential equation. Afterward, the research on the Brown Motion came into the fractional Brown Motion Where Bl is a standard Brown MotionΦis a Borel measurable function satisfying certain conditions.In 1995, Cranston and Lejan extended this model and introduced the self-attracting diffusion. According to the properties of fractional Brownian motion, Yan and others considered the former function which is drove by fractional Brownian motion BH and called its solution fractional self-attracting diffusion. Meanwhile they have made detailed exposition about a special situation-the linear fractional self-attracting diffusion.We first consider how to get the get the discrete version of the linear fractional self-attracting diffusion driven by fractional Brown Motion. Because the shape change of the polymer is discrete, a discrete version is better for the use. Making use of the fractional Brown Motion and the Donsker theorem, we get the discrete version of the linear fractional self-attracting diffusion as follow: What's more, we will explicitly prove the convergence and the weak convergence of former sequenceSecondly, we estimate the parameter a. There is a parameter of a in the linear fractional self-attracting diffusion:This brings us great inconvenience to the research and the use. Firstly we use the maximum likelihood method to obtain maximum likelihood estimator:What's more, we will explicitly prove the unbiasedness, consistency and the central limit theorem of the maximum likelihood estimator.Besides, we also use the least squares estimate method and get the least squares estimator:At last, we prove the asymptotic behavior of the least squares estimator.