Recursive identification in continuous-time stochastic processes

Recursive parameter estimation in diffusion processes is considered. First, stability and asymptotic properties of the global, off-line MLE (maximum likelihood estimator) are obtained under explicit conditions. The MLE evolution equation is then derived by employing a generalized Itô differentiation rule. This equation, which is highly sensitive to initial conditions, is then modified to yield an algorithm (infinite dimensional in general) which results in an estimator that, irrespective of initial conditions, is consistent and asymptotically efficient and in addition, converges rapidly to the MLE. The structure of the algorithm indicates that well known gradient and Newton type algorithms are first-order approximations. The results cover a wide class of processes, including nonstationary or even divergent ones.