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Register a new userMultilevel Particle Filters for Levy-driven stochastic differential equations
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Kody Law
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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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 $O(epsilon^{-3})$.
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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 with a high order scheme for the Brownian driven component, applied between the jump times. The overall approximation is analyzed using a stochastic splitting argument. The resulting error bound involves separate contributions of the compound Poisson approximation and of the discretization scheme for the Brownian part, and allows, on one hand, to balance the two contributions in order to minimize the computational time, and on the other hand, to study the optimal design of the approximating compound Poisson process. For driving processes whose Levy measure explodes near zero in a regularly varying way, this procedure allows to construct discretization schemes with arbitrary order of convergence.
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