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This article analyzes and compares two general techniques of rare eventsimulation for generating paths of Markov processes over fixed time horizons:backtracking and exponential tilting. These two methods allow to compute theprobability that the process ends within a rare region, which is unlikely to beattained. Backtracking consists in reversing the time of the process: the pathis obtained backwards, from the terminal point until the initial one. Theterminal point is generated from an appropriately chosen distribution thatcovers well the arrival region. Exponential tilting is a general technique forobtaining an alternative sampling probability measure, under which the processis likely to hit the rare region at terminal time. We show that both methodsbelong to the same class of importance sampling procedures, by providing thecommon mathematical framework of these two conceptually different methods ofsampling rare trajectories. Besides this analytical comparison, we compare thetwo methods numerically, by means of a simple random walk and a process withmeta-stable states. The numerical analysis shows that both methods possessdistinct areas of application where they exhibit greater efficiency. Detailedalgorithms of the proposed simulation methods are provided.

A Unified Perspective on Sampling Algorithms for Rare Trajectories of Discrete Markov Processes