In this article, we consider the estimation problem of continuous-time stochastic systems with discrete measurements, having linear drift and nonlinear diffusion term. We build the infinite-dimensional linear system equivalent to this class of systems by means of a Carleman linearization approach. Based on this embedding, we investigate the properties of the moment equations of the original system, and we show that it is possible to write the optimal linear filter, for which a finite-dimensional approximation can be implemented. We validate the approach by showing that the resulting algorithm may outperform widely used continuous-discrete filters without increasing the computational burden.