| 随机微分方程的几种参数估计方法 |
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| 作者姓名： | 蔡昕芮 王丽瑾 |
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| 作者单位： | 中国科学院大学数学科学学院, 北京 100049 |
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| 基金项目： | Supported by the National Natural Science Foundation of China (11471310,11071252) |
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| 摘 要： | 提出3种基于离散观测数据的随机微分方程参数估计的方法。第1种方法应用于线性随机微分方程。推导出这类方程的真解的相关运算服从的分布，使观测数据的运算也服从此分布，由此来估计漂移系数与扩散系数中的未知参数。第2种方法用于Itô型随机微分方程。推导出Euler-Maruyama格式的数值解的相关运算服从的分布，使观测数据的运算服从此分布，由此来估计参数。第3种方法用于Stratonovich型随机微分方程。推导出中点格式的数值解的相关运算服从的分布，使观测数据的运算服从此分布，以此来估计参数。数值实验验证了这3种方法的有效性。数值实验显示，Euler-Maruyama格式参数估计的误差约为O（h0.5）阶，中点格式参数估计的误差约为O（h）阶，其中h是数值方法的时间步长。我们提出的3种估计方法均比文献中已有的EM-MLE方法更精确。
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| 关 键 词： | 随机微分方程 参数估计 Euler-Maruyama 格式 中点格式 |
| 收稿时间： | 2016-03-17 |
| 修稿时间： | 2016-06-12 |
Some methods of parameter estimation for stochastic differential equations |
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| Authors: | CAI Xinrui WANG Lijin |
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| Institution: | School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China |
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| Abstract: | We propose three methods of parameter estimation based on discrete observation data for stochastic differential equations (SDEs). The first method is designed for linear stochastic differential equations (SDEs). For these equations we deduce distribution of certain operation of the exact solution and assume that the relevant operation of the observed data obey this distribution, from which we estimate the unknown parameters in the drift and diffusion coefficients. In the second method, we suppose that certain operation of the observation data and that of the numerical solution arising from the Euler-Maruyama scheme for the SDEs of Itô sense obey the same distribution, from which the unknown parameters can be estimated. We use the third method for SDEs of Stratonovich sense. For these equations we derive the distribution of relevant operation of the numerical solution produced by the midpoint scheme and let the same operation of the data obey this distribution to get estimation of the unknown parameters. Numerical results show validity of the proposed methods, and illustrate that the estimation error produced by the Euler-Maruyama scheme is about of order O(h0.5) while that by the midpoint scheme is about of order O(h), with h being the time step size of the numerical methods. Furthermore, the numerical results show that our methods are more accurate than the existing EM-MLE estimator. |
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| Keywords: | stochastic differential equations parameter estimation Euler-Maruyama scheme midpoint scheme |
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