Multilevel particle filters for Lévy-driven stochastic differential equations

We develop algorithms for computing expectations with respect to the laws of models associated to stochastic differential equations driven by pure Lévy processes. We consider filtering such processes as well as pricing of path dependent options. We propose a multilevel particle filter to address the computational issues involved in solving these continuum problems. We show via numerical simulations and theoretical results that under suitable assumptions regarding the discretization of the underlying driving Lévy proccess, the cost to obtain MSE $$\mathcal {O}(\epsilon ^2)$$O(ϵ2) scales like $$\mathcal {O}(\epsilon ^{-2})$$O(ϵ-2) for our method, as compared with the standard particle filter $$\mathcal {O}(\epsilon ^{-3})$$O(ϵ-3).