Robust M-estimators in diffusion processes

Methods of robust estimation in diffusion processes are given by means of M-estimation. It is shown that the asymptotic variance of an M-estimator is obtained by applying a certain integral operator to the influence function and integrating its square. Under the condition of boundedness of the influence function, the existence of an optimal robust M-estimator is shown and an approximately optimal practical method is given. Moreover, as another criterion of robustness we consider a norm of integral type and show that the corresponding optimal robust M-estimator is obtained by solving a boundary value problem of a second order differential equation. Finally, as an illustrative example the Ornstein-Uhlenbeck process is discussed.