Unified perspective on exponential tilt and bridge algorithms for rare trajectories of discrete Markov processes

This article analyzes and compares two general techniques of rare event simulation for generating paths of Markov processes over fixed time horizons: exponential tilting and stochastic bridge. These two methods allow us to accurately compute the probability that a Markov process ends within a rare region which is unlikely to be attained. Exponential tilting is a general technique for obtaining an alternative or tilted sampling probability measure, under which the Markov process becomes likely to hit the rare region at terminal time. The stochastic bridge technique involves conditioning paths towards two endpoints: the terminal point and the initial one. The terminal point is generated from some appropriately chosen probability distribution that covers well the rare region. We show that both methods belong to the class of importance sampling procedures by providing a common mathematical framework of these two conceptually different methods of sampling rare trajectories. We also conduct a numerical comparison of these two methods, revealing distinct areas of application for each Monte Carlo method, where they exhibit superior efficiency. Detailed simulation algorithms are provided.