In this article we introduce two new estimates of the normalizing constant (or marginal likelihood) for partially observed diffusion (POD) processes, with discrete observations. One estimate is biased but non-negative and the other is unbiased but not almost surely non-negative. Our method uses the multilevel particle filter of [11]. We show that, under assumptions, for Euler discretized PODs and a given ε > 0 in order to obtain a mean square error (MSE) of O(ε2) one requires a work of O(ε−2.5) for our new estimates versus a standard particle filter that requires a work of O(ε−3). Our theoretical results are supported by numerical simulations.
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