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Two-scale finite element discretizations for infinitesimal generators of jump processes in finance

by F. Liu and N. Reich and A. Zhou

(Report number 2008-23)

Abstract
We propose and analyze a number of two-scale discretization schemes for infinitesimal generators of jump processes arising in Finance. It is shown that the number of degrees of freedom of the the two-scale discretization is significantly smaller than that of the standard one-scale finite element approach while at the same time preserving the accuracy of the one-scale discretization. Based on the two-scale discretizations, we present some local and parallel algorithms. The main idea of these algorithms is to use a coarse grid to approximate the low frequencies and then to use a fine grid to correct the relatively high frequencies by some local/parallel procedures. In particular, we present two so-called combination based two-scale finite element discretization schemes on tensor product domains. They can be carried out in parallel. As a result, both the computational time and the storage can be reduced considerably. Finally, we illustrate how a combination of wavelet and Lagrangian finite element basis functions can be applied to further reduce the complexity arising from the non-locality of the integro-differential operators under consideration.

Keywords:

@Techreport{LRZ08_387,
  author = {F. Liu and N. Reich and A. Zhou},
  title = {Two-scale finite element discretizations for infinitesimal generators of jump processes in finance},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2008-23},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2008/2008-23.pdf },
  year = {2008}
}

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