离散观测扩散过程参数极大似然估计的高效算法研究

Diffusion process model governed by stochastic differential equations(SDE) has been widely used to model systems of finance, biology, etc. However, the problem of parameter estimation of SDE models based on the measurements of output variables, has not been well explored using modern statistical and numerical methods. This project will mainly construct new and high-efficiency algorithm to estimate the parameter of general nonlinear stochastic system and stochastic delay system models. For general nonlinear stochastic system models, if the observations are measured without measurement error, we will consider a high-order difference method to numerically solve the corresponding nonlinear parabolic equations when 1-dimensional SDEs are considered, so as to obtain the likelihood function. And an improved local linearizaion method with high-order will be constructed when multi-dimensional SDEs are considered. If the observations are measured with measurement error, the improved Kalman filter method will be constructed to obtain the likelihood functions.For stochastic delay system models, if the observations are measured without measurement error, we will consider a high-order difference method to numerically solve the corresponding parabolic equations with delay when 1-dimensional SDEs are considered, so as to obtain the likelihood function. If the observations are measured with measurement error, the improved Kalman filter method will be constructed to obtain the likelihood functions. Thus, the parameter of the considered stochastic system models can be estimated by maximum likelihood estimation method. Finally, the considered models are used to model the data of finance and biology.

由随机微分方程(SDE)所控制的扩散过程模型已广泛应用于金融、生物等领域，目前如何利用已知状态变量的离散观测值，估计SDE模型中的未知参数，还有待于利用现代统计方法进行深入的探索和研究。本项目主要研究如何构造新的高效极大似然算法估计一般非线性随机系统和随机延迟系统模型中的未知参数问题。对于一般非线性随机系统模型，当观测值不带测量误差时，一维情形下，拟构造高阶差分算法求解相应的非线性抛物型方程，获得似然函数；多维情形下，拟构造改进局部线性化方法获得似然函数；当观测值带测量误差时，拟构造更高精度的改进滤波法获得似然函数。对于随机延迟系统模型，当观测值不带测量误差时，一维情形下，拟构造高阶差分算法求解相应的延迟抛物型方程，获得似然函数；当观测值带测量误差时，拟构造改进的滤波法获得似然函数。根据得到的似然函数对所考察随机系统模型进行参数估计，并考虑这些模型在拟合金融、生物等实际数据时的应用。

由随机微分方程(SDE)所控制的扩散过程模型已广泛应用于金融、生物等领域，目前如何利用已知状态变量的离散观测值，估计SDE模型中的未知参数，还有待于利用现代统计方法进行深入的探索和研究。本项目主要研究如何构造新的高效极大似然算法估计一般非线性随机系统，随机延迟系统模型以及分数阶随机微分方程中的未知参数问题。对于一般非线性随机系统模型，当观测值不带测量误差时，一维情形下，拟构造高阶差分算法求解相应的非线性抛物型方程，获得似然函数；当观测值带测量误差时，拟构造更高精度的改进滤波法获得似然函数。对于随机延迟系统模型，当观测值不带测量误差时，一维情形下，拟构造高阶差分算法求解相应的延迟抛物型方程，获得似然函数；对于分数阶随机系统模型，当观测值不带测量误差时，一维情形下，拟构造高阶差分算法求解相应的分数阶抛物型方程，获得似然函数。根据得到的似然函数对所考察随机系统模型进行参数估计，并考虑这些模型在拟合金融、生物等实际数据时的应用。

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