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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 $O(epsilon^{-3})$.
We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Levy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Levy process w
We construct an efficient integrator for stochastic differential systems driven by Levy processes. An efficient integrator is a strong approximation that is more accurate than the corresponding stochastic Taylor approximation, to all orders and indep
We develop and analyze a method, density tracking by quadrature (DTQ), to compute the probability density function of the solution of a stochastic differential equation. The derivation of the method begins with the discretization in time of the stoch
In this paper, we study almost periodic solutions for semilinear stochastic differential equations driven by L{e}vy noise with exponential dichotomy property. Under suitable conditions on the coefficients, we obtain the existence and uniqueness of bo
In this paper, we use a unified framework to study Poisson stable (including stationary, periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-peri