We address the statistical filtering problem in dynamical models with jumps. When a particular application is adequately modeled by linear and Gaussian probability density functions with jumps, a usual method consists in approximating the optimal Bayesian estimate (in the sense of the Minimum Mean Square Error (MMSE)) in a linear and Gaussian Jump Markov State Space System (JMSS). Practical solutions include algorithms based on numerical approximations or on Sequential Monte Carlo (SMC) methods. In this paper, we propose a class of alternative methods which consists in building statistical models which, locally, similarly model the problem of interest, but in which the computation of the MMSE estimate can be computed exactly (without numerical nor SMC approximations) and at a computational cost which is linear in the number of observations