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We consider continuous-time models where the observed process depends on an unobserved jump Markov Process. We develop a sequential Monte Carlo algorithm which makes it possible to filter and smooth this latent process, and compute the likelihood pointwise. We develop a Rao-Blackwellisation technique which allows to significantly reduce the Monte Carlo noise of this algorithm. Possible extensions of our algorithm and further directions of research are discussed. * We are grateful to Tobyasdoi:10.1051/proc:071903 fatcat:syncrqnwuvf3dkxwhtr5xftuau