A novel method and comparison of methods for constructing Markov bridges

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.