$M$-estimation for discretely sampled diffusions

We study the estimation of a parameter in the nonlinear drift coefficient of a stationary ergodic diffusion process satisfying a homogeneous Itô stochastic differential equation based on discrete observations of the process, when the true model does not necessarily belong to the observer's model. Local asymptotic normality of $M$-ratio random fields are studied. Asymptotic normality of approximate $M$-estimators based on the Itô and Fisk–Stratonovich approximations of a continuous $M$-functional are obtained under a moderately increasing experimental design condition through the weak convergence of approximate $M$-ratio random fields. The derivatives of an approximate log-$M$ functional based on the Itô approximation are martingales, but the derivatives of a log-$M$ functional based on the Fisk–Stratonovich approximation are not martingales, but the average of forward and backward martingales. The averaged forward and backward martingale approximations have a faster rate of convergence than the forward martingale approximations.