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，我们探讨了基于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。主要目标在于精确估计马尔可夫跳跃过程的无穷小生成元，这是众多应用中的关键任务。为解决该问题，我们首先回顾了已知的从条件端点生成的马尔可夫跳跃过程样本路径（即马尔可夫桥）的构建方法。同时，我们提出了一种基于时间反转概念的新算法，这是本研究的主要贡献。随后，将所提出的方法应用于估计马尔可夫跳跃过程的无穷小生成元。为此，我们结合了马尔可夫链蒙特卡洛技术与蒙特卡洛期望最大化算法。实验结果表明，该方法能有效提供准确的参数估计。为评估其性能，我们与现有技术（包括二分法、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，重点关注计算速度与精度。值得注意的是，本方法在保持高精度的同时，在所有备选方案中具有最快的运算速度。