离散观测下线性自吸引扩散的最小二乘估计

Least Squares Estimation for Self-Attracting Diffusion with Discrete Observations

摘要: 考虑如下分数布朗运动驱动的自吸引扩散过程X_t^H=B_t^H-θ∫₀^t∫₀^t（X_s^H-X_u^H）duds+vt，其中B_t^H表示Hurst指数为H∈[1/2，1）的分数布朗运动，而θ>0， v∈ℝ为未知参数。在离散观测下，给出了这两个未知参量的最小二乘估计量和，验证了它们无相合性同时构造新的弱相合估计量。

Abstract: In this paper, the self-attracting diffusion process driven by fractional Brownian motion X_t^H=B_t^H-θ∫₀^t∫₀^t（X_s^H-X_u^H）duds+vt is considered, where B_t^H is fractional Brownian motion with Hurst index H∈[1/2，1）, and θ>0， v∈ℝ are two unknown parameters. With discrete observation, we research the least squares estimators and for the unknown parameters. It is proved that they are not weakly consistency and we also construct some new estimators which have weakly consistency.