带有α-稳定OU噪声小干扰的随机过程的参数与非参数估计

近年来，对带有小干扰项随机微分方程未知参数估计问题的研究一直是国内外众多学者研究的热点。本项目主要研究带有α-稳定OU噪声小干扰的随机过程的参数与非参数估计问题。首先，利用Fatou引理、Lévy过程理论等方法探讨带有α-稳定OU噪声小干扰的随机微分方程解的存在唯一性和p(0<p<α)阶矩有界性。其次，使用参数估计方法、Markov不等式以及α-稳定过程性质分别讨论离散和连续两种时间观测情形下带有α-稳定OU噪声小干扰的随机过程参数估计量的相合性、收敛速率以及渐近分布特征。最后，借助非参数估计方法、α-稳定过程理论、鞅不等式分别研究离散、连续时间观测情形下带有α-稳定OU噪声小干扰的随机过程非参数估计量的相合性、收敛速率以及渐近分布。本项目的研究成果将进一步丰富带有小干扰项随机微分方程的统计推断理论。

In recent years, the research on the estimation of unknown parameters for stochastic differential equations with small disturbance has been the hot issue of scholars at home and abroad. The aim of the project is to study the parametric and nonparametric estimation for stochastic processes with small α-stable Ornstein-Uhlenbeck noises. The research content is as follows. Firstly, the existence and uniqueness of the solutions p-moment (0<p<α) boundedness for stochastic differential equations with small α-stable Ornstein-Uhlenbeck noises are investigated by using the Fatou's lemma, Lévy processes theory and so on. Secondly, the consistency, rate of convergence and asymptotic distribution of parametric estimator for stochastic processes with small α-stable Ornstein-Uhlenbeck noises are discussed by applying parameter estimation methods, Markov’s inequality, and properties of α-stable processes from discrete-time observation and continuous-time observation, respectively. Finally, the consistency, rate of convergence and asymptotic distribution of non-parametric estimator for stochastic processes with small α-stable Ornstein-Uhlenbeck noises are analyzed by applying non-parametric estimation method, theory of α-stable processes and martingale's inequalities based on discrete-time observation and continuous-time observation, respectively. The research results of the project will further enrich the theory of statistical inference for stochastic differential equations.