Recursive identification of continuous-time linear stochastic systems - Convergence w.p.1 and in Lq

We present a convergence theorem for a computable continuous-time recursive maximum likelihood method with resetting, under realistic conditions. Resetting takes place if the estimator process hits the boundary of a pre-specified compact domain, or if the rate of change, in a stochastic sense, of the parameter process would hit a fixed threshold. The modified recursive maximum likelihood estimator converges to the true value of the parameter almost surely and in L_q for any q, provided that the threshold imposed on the rate of change of the parameter is sufficiently small. We also show that the rate of convergence in L_q is O(T^-1/2). The proof, the outline of which will be given, is based on an extension of the scheme of Benveniste, Metivier and Priouret (BMP) to estimation problems described in terms of continuous-time linear stochastic systems.