Approximate maximum likelihood estimation for one‐dimensional diffusions observed on a fine grid
针对细网格观测的一维随机微分方程，提出一种基于Itô-Taylor展开的转移密度近似方法，得到闭式似然函数，进而计算近似极大似然估计，理论分析和数值模拟表明其精度和速度优于其他近似方法。
Abstract 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 ‐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.