Maximum likelihood estimation of time-inhomogeneous diffusions

We extend the maximum likelihood estimation method of Ait-Sahalia (Econometrica 70 (2002) 223) for time-homogeneous diffusions to time-inhomogeneous ones. We derive a closed-form approximation of the likelihood function for discretely sampled time-inhomogeneous diffusions, and prove that this approximation converges to the true likelihood function and yields consistent parameter estimates. Monte Carlo simulations for several financial models reveal that our method largely outperforms other widely used numerical procedures in approximating the likelihood function. Furthermore, parameter estimates produced by our method are very close to the parameter estimates obtained by maximizing the true likelihood function, and superior to estimates obtained from the Euler approximation.