We consider a one-dimensional stochastic differential equation that is observed on a fine grid of equally spaced time points. A novel approach for approximating the transition density of the stochastic differential equation is presented, which is based on an Itô-Taylor expansion of the sample path, combined with an application of the so-called $ϵ$-expansion. The resulting approximation is economical with respect to the number of terms needed to achieve a given level of accuracy in a high-frequency sampling framework. This method of density approximation leads to a closed-form approximate likelihood function from which an approximate maximum likelihood estimator may be calculated numerically. A detailed theoretical analysis of the proposed estimator is provided and it is shown that it compares favorably to the Gaussian likelihood-based estimator and does an excellent job of approximating the exact, but usually intractable, maximum likelihood estimator. Numerical simulations indicate that the exact and our approximate maximum likelihood estimator tend to be close, and the latter performs very well relative to other approximate methods in the literature in terms of speed, accuracy, and ease of implementation.

中文翻译：

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在精细网格上观察到的一维扩散的近似最大似然估计

我们考虑在等间隔时间点的精细网格上观察到的一维随机微分方程。提出了一种近似随机微分方程过渡密度的新方法，该方法基于样本路径的 Itô-Taylor 展开，并结合所谓的$\epsilon$-扩张。就在高频采样框架中实现给定精度水平所需的项数而言，所得近似值是经济的。这种密度近似方法产生一个封闭形式的近似似然函数，可以从该函数以数值方式计算近似最大似然估计量。对所提出的估计器进行了详细的理论分析，结果表明它优于基于高斯似然的估计器，并且在逼近精确但通常难以处理的最大似然估计器方面做得很好。数值模拟表明，精确的最大似然估计和我们的近似最大似然估计趋于接近，并且后者在速度、准确性方面相对于文献中的其他近似方法表现得非常好，