Approximate maximum likelihood estimation for one-dimensional diffusions observed on a fine grid

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.