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Optimal simulation schemes for Lévy driven stochastic differential equations


Authors: Arturo Kohatsu-Higa, Salvador Ortiz-Latorre and Peter Tankov
Journal: Math. Comp. 83 (2014), 2293-2324
MSC (2010): Primary 65C30, 60G51
DOI: https://doi.org/10.1090/S0025-5718-2013-02786-X
Published electronically: December 17, 2013
MathSciNet review: 3223333
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Abstract: We consider a general class of high order weak approximation schemes for stochastic differential equations driven by Lévy processes with infinite activity. These schemes combine a compound Poisson approximation for the jump part of the Lévy 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 Lévy measure explodes near zero in a regularly varying way, this procedure allows us to construct discretization schemes with arbitrary order of convergence for sufficiently regular functionals.


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Additional Information

Arturo Kohatsu-Higa
Affiliation: Department of Mathematical Sciences and Japan Science and Technology Agency, Ritsumeikan University
Email: arturokohatsu@gmail.com

Salvador Ortiz-Latorre
Affiliation: Department of Mathematics, Centre of Mathematics for Applications, University of Oslo
Email: salvador.ortiz-latorre@cma.uio.no

Peter Tankov
Affiliation: LPMA, Univeristy Paris Diderot — Paris 7
Email: tankov@math.univ-paris-diderot.fr

DOI: https://doi.org/10.1090/S0025-5718-2013-02786-X
Keywords: L\'evy-driven stochastic differential equations, high order discretization schemes, weak approximation, regular variation
Received by editor(s): April 22, 2012
Received by editor(s) in revised form: November 30, 2012, and January 4, 2013
Published electronically: December 17, 2013
Additional Notes: The first author acknowledges financial support from the Japanese goverment
The second author acknowledges financial support from the Catalan government, BP-DGR 2009.
Article copyright: © Copyright 2013 American Mathematical Society