We develop algorithms for computing expectations of the laws of models associated to stochastic differential equations (SDEs) driven by pure Levy processes. We consider filtering such processes and well as pricing of path dependent options. We propose a multilevel particle filter (MLPF) to address the computational issues involved in solving these continuum problems. We show via numerical simulations and theoretical results that under suitable assumptions of the discretization of the underlying driving Levy proccess, our proposed method achieves optimal convergence rates. The cost to obtain MSE $O(epsilon^2)$ scales like $O(epsilon^{-2})$ for our method, as compared with the standard particle filter $O(epsilon^{-3})$.
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