Unbiased simulation of rare events in continuous time

For rare events described in terms of Markov processes, truly unbiased estimation of the rare event probability generally requires the avoidance of numerical approximations of the Markov process. Recent work in the exact and $\varepsilon$-strong simulation of diffusions, which can be used to almost surely constrain sample paths to a given tolerance, suggests one way to do this. We specify how such algorithms can be combined with the classical multilevel splitting method for rare event simulation. This provides unbiased estimations of the probability in question. We discuss the practical feasibility of the algorithm with reference to existing $\varepsilon$-strong methods and provide proof-of-concept numerical examples.