$\mathcal{L}_1$-optimal filtering of Markov jump processes. I. Exact solution and numerical implementation schemes

Part I of this research work is devoted to the development of a class of numerical solution algorithms for the filtering problem of Markov jump processes by indirect continuous-time observations corrupted by Wiener noises. The expected $\mathcal{L}_1$ norm of the estimation error is chosen as an optimality criterion. The noise intensity depends on the state being estimated. The numerical solution algorithms involve not the original continuous-time observations, but the ones discretized by time. A feature of the proposed algorithms is that they take into account the probability of several jumps in the estimated state on the time interval of discretization. The main results are the statements on the accuracy of the approximate solution of the filtering problem, depending on the number of jumps taken into account for the estimated state, on the discretization step, and on the numerical integration scheme applied. These statements provide a theoretical basis for the subsequent analysis of particular numerical schemes to implement the solution of the filtering problem.