Tracking of signals and its derivatives in Gaussian white noise (Q1275949)

Let \(S(t)\) be a smooth signal, the information of which is given by the observation of a random process \(X_t,\;t\geq 0\), with \[ dX_t=S(t)dt+\varepsilon dW_t,\quad X_0=0, \] where \(W_t,\;t\geq 0\), is a Wiener process and \(\varepsilon \) is a small parameter. An on-line tracking algorithm for the signal and its derivatives up to a given order is proposed and the asymptotic optimality in the minimax sense, with respect to small intensity of the noise, is established. The signal \(S\) and its respective derivatives are assumed to be Hölder continuous and bounded at zero.