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This note deals with stochastic continuous-discrete state-space models, that is stochastic differential systems with sampled discrete measurements. The filtering problem is investigated, with the purpose to provide the state estimate at the samples times. The general setting of a nonlinear drift and of a nonlinear multiplicative noise is considered, as well as of a nonlinear state-to-output function. According to the spirit of the Extended Kalman Filter, the original nonlinear differential system is linearized and discretized; then a bilinear system in the discrete-time framework is obtained, and the minimum variance filter equations are written. The novelty of the paper consists in the use of a state observer for nonlinear differential systems that provides the prediction to the filter equations and also the point around which the linear approximation is achieved. The observer equations make use of a modified version of a class of observers for nonlinear differential systems, coping with the problem of the discrete feature of the measurements, by modeling them as continuous measurements affected by a time-varying delay. Such an Observer Follower Filter approach has been recently applied to stochastic (purely) continuous-time framework. Numerical results show the good performances of the proposed approach with respect to the standard methodologies.