分数布朗运动驱动的随机微分方程的参数估计问题

Parameter Estimation of Stochastic Differential Equations Driven by Fractional Brownian Motion

考虑由分数布朗运动驱使的随机微分方程的参数估计问题,首先,通过一定的方法将其转化为一个半鞅过程,再对其做极大似然估计,并证明估计量的渐近一致性和渐近分布;其次,通过一阶随机微分方程的Bernstein-von Mises定理,得到参数的后验函数的渐近分布,进一步通过贝叶斯决策得到参数的最优估计——贝叶斯估计,并证明贝叶斯估计量的渐近性质.

Considering the parameter estimation problem of stochastic differential equation with fractional Brownian motion.Firstly,we transform the stochastic process into a semi-martingale process by some methods.Then we solve its maximum likelihood estimator and prove asymptotic consistency and asymptotic normality of the estimator.Secondly,by the Bernstein-von Mises theorem for first order stochastic differential equations,we get the asymptotic distribution of posterior function of parameter.Further,the optimal estimation of parameters is acquired by Bayesian decision,which is Bayesian estimation.After,we prove asymptotic properties of the Bayesian estimator.