In this article, we consider the finite horizon filtering problem of discrete-time Markov jump systems (MJS). In the first part, we consider an MJS satisfying some general nonlinear conditions. It is obtained, for a fixed set of auxiliary constants, filter gains, based on a set of coupled Riccati-like difference equations, that yield a minimum upper bound for the estimation error covariance matrix. When the nonlinear parameters are set to zero, the MJS becomes a Markov jump linear system (MJLS) and, in the second part of the article, it is shown that the obtained filter gains derived from a set of coupled Riccati-like difference equations provide the optimal “prediction-correction” Markovian filter for the nominal MJLS (which reduces to the standard Kalman filter for the no jump case). The article is concluded with some numerical examples.