A Comparative Study of Likelihood Approximations for Univariate Diffusions
比较了Hermite多项式展开逼近一维扩散过程转移密度的准确性，发现处理不可约扩散时存在积分困难，引入高斯求积法解决了问题并提高了可靠性，模拟和VIX数据实证验证了方法有效性。
Abstract Maximum likelihood estimation of the parameters of stochastic differential equations commonly used in finance requires numerical approximation of their transitional probability density functions. This article undertakes a comparative study of the accuracy of Hermite polynomial expansion approximations for univariate diffusions and checks how the accuracy of the existing methods responds to increasing the order of the approximation. It is found that one class of expansion dealing with irreducible diffusions is particularly problematic due to the need to evaluate a number of troublesome integrals. A Gaussian quadrature is introduced which resolves the problem and improves the reliability of the expansion. A simulation study demonstrates all the methods in action and provides insight into the practical aspects of using these expansions. An empirical application using data on the VIX indicates that the proposed method based on the Gaussian quadrature performs very well when applied to financial data.