A continuous-time Markov chain rate change formula for simulation, model selection, filtering and theory is proven. It is used to develop Markov chain importance sampling, rejection sampling, branching particle filtering algorithms and filtering equations akin to the Duncan-Mortensen-Zakai equation and the Fujisaki-Kallianpur-Kunita equation but for Markov signals with general continuous-time Markov chain observations. A direct method of solving these filtering equations is given that, for example, applies to trend, volatility and/or parameter estimation in financial models given tick-by-tick market data. All the results also apply to continuous-time Hidden Markov Models (CTHMM), which have become important in applications like disease progression tracking, as special cases and the corresponding CTHMM results are stated as corollaries.

翻译：本文证明了一个适用于模拟、模型选择、滤波及理论的连续时间马尔可夫链速率变换公式。利用该公式，我们发展了马尔可夫链重要性采样、拒绝采样、分支粒子滤波算法，以及类似于Duncan-Mortensen-Zakai方程和Fujisaki-Kallianpur-Kunita方程但针对一般连续时间马尔可夫链观测下马尔可夫信号的滤波方程。本文给出了一种直接求解这些滤波方程的方法，该方法可应用于例如基于逐笔市场数据的金融模型中的趋势、波动率和/或参数估计。所有结果同样适用于连续时间隐马尔可夫模型（CTHMM）——该模型在疾病进展追踪等应用中已变得至关重要，对于这些特例，相应的CTHMM结果以推论形式给出。