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We present a sequential Monte Carlo algorithm for Markov chain trajectorieswith proposals constructed in reverse time, which is advantageous when pathsare conditioned to end in a rare set. The reverse time proposal distribution isconstructed by approximating the ratio of Green's functions in Nagasawa'sformula. Conditioning arguments can be used to interpret these ratios aslow-dimensional conditional sampling distributions of some coordinates of theprocess given the others. Hence the difficulty in designing SMC proposals inhigh dimension is greatly reduced. We illustrate our method on estimating anoverflow probability in a queueing model, the probability that a diffusionfollows a narrowing corridor, and the initial location of an infection in anepidemic model on a network.

Inference and rare event simulation for stopped Markov processes via reverse-time sequential Monte Carlo