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.

这项研究工作的第一部分致力于通过间接连续时间观测被维纳噪声破坏的马尔可夫跳跃过程的滤波问题，开发一类数值解算法。预期的\（{{\ mathcal {L}}} _ {1} \）选择估计误差的范数作为最优准则。噪声强度取决于估计的状态。数值解算法不涉及原始的连续时间观测值，而是涉及时间离散的观测值。所提出算法的特征在于，它们考虑了离散化时间间隔上估计状态中几次跳跃的可能性。主要结果是关于滤波问题的近似解的准确性的陈述，这取决于为估计状态考虑的跳跃次数，离散化步骤以及所应用的数值积分方案。这些陈述为后续特定数值方案的分析提供理论基础，以实现过滤问题的解决方案。