In this study, we address the central issue of statistical inference for Markov jump processes using discrete time observations. The primary problem at hand is to accurately estimate the infinitesimal generator of a Markov jump process, a critical task in various applications. To tackle this problem, we begin by reviewing established methods for generating sample paths from a Markov jump process conditioned to endpoints, known as Markov bridges. Additionally, we introduce a novel algorithm grounded in the concept of time-reversal, which serves as our main contribution. Our proposed method is then employed to estimate the infinitesimal generator of a Markov jump process. To achieve this, we use a combination of Markov Chain Monte Carlo techniques and the Monte Carlo Expectation-Maximization algorithm. The results obtained from our approach demonstrate its effectiveness in providing accurate parameter estimates. To assess the efficacy of our proposed method, we conduct a comprehensive comparative analysis with existing techniques (Bisection, Uniformization, Direct, Rejection, and Modified Rejection), taking into consideration both speed and accuracy. Notably, our method stands out as the fastest among the alternatives while maintaining high levels of precision.

翻译：在本研究中，我们解决了利用离散时间观测对马尔可夫跳跃过程进行统计推断的核心问题。当前的主要任务是准确估计马尔可夫跳跃过程的无穷小生成元，这在各种应用中至关重要。为解决此问题，我们首先回顾了从给定端点条件（称为马尔可夫桥）的马尔可夫跳跃过程生成样本路径的现有方法。此外，我们引入了一种基于时间反转概念的新算法，这构成了我们的主要贡献。随后，我们采用所提出的方法来估计马尔可夫跳跃过程的无穷小生成元。为实现这一目标，我们结合使用了马尔可夫链蒙特卡洛技术和蒙特卡洛期望最大化算法。我们方法获得的结果证明了其在提供准确参数估计方面的有效性。为评估所提方法的效能，我们与现有技术（二分法、均匀化法、直接法、拒绝法及改进拒绝法）进行了全面的比较分析，并综合考虑了速度与精度。值得注意的是，我们的方法在保持高精度的同时，在所有备选方案中速度最快。