Nonparametric Estimation Of Linear Multiplier For Stochastic Differential Equations Driven By α-Stable Noise

This paper deals with the problem of nonparametric estimation of the linear multiplier for Ornstein-Uhlenbeck process（OU process）with a generalized Lévy noises:Lt=aWt+bZt,Wt is a standard Wiener process and Zt is a strictlyα-stable Lévy process,a and b are known constants,based on continuous-time observation.Under some certain conditions,we derive the uniform consistency and the rate of convergence of the nonparametric estimator,and the asymptotic distribution ofθ（t）under different conditions is obtained.When a=0,the asymptotic distribution of θ（t）is strictly α-stable,otherwise the asymptotic distribution is Gaussian,and more general conclusions are obtained:for OU processes with multiple different α-stable noises,the asymptotic distribution of the estimator constructed in this paper will be determined by the largest parameter α in those noise.Compared with other studies on similar problems,the kernel-type estimator presented in this paper not only guarantees the smoothness and existence,but also guarantees that the parameters in the asymptotic distribution are constant,which is an advantage of the results in this paper while the asymptotic distribution of parameters in other studies on linear multiplier parameters is the joint distribution related to xt.