由双分数布朗运动驱动的线性自排斥扩散的最小二乘估计

Least square estimation for a self-repelling diffusion process driven by bi-fractional Brownian motion

考虑如下双分数线性自排斥扩散过程:X^H,K=Bt^H,K+θ∫^t0∫^s0(Xs^H,K-X^H,Ku)duds,这里X0H,K=0,θ>0是未知参数,BH,K是Hurst参数H与K满足H∈(0,1),K∈(0,1),1/2≤HK<1的双分数布朗运动。该过程为一类自交互扩散过程的类似过程(参见文献[1-2])。论文研究目的是在连续观测条件下研究参数θ的最小二乘估计问题,并给出它的渐近分布。

In this paper,we consider the following self-repelling diffusion process driven by bi-fractional Brownian motion:X^H,K=Bt^H,K+θ∫^t0∫^s0(Xs^H,K-X^H,Ku)duds.Here,X^H,K0=0,0>0 is an unknown parameter,and BH,K is a bifractional Brownian motion with Hurst parameters H and K satisfying H∈(0,1),K∈(0,1],1/2 ≤HK<1.The process is an analogue of the self-attracting diffusion(Ref.[1-2]).We have studied the asymptotic behaviors of the least squares estimator of θ under the continuous observation.