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On Kolmogorov equations for anisotropic multivariate Lévy processes

Reich, N. ; Schwab, C. ; Winter, C.

In: Finance and Stochastics, 2010, vol. 14, no. 4, p. 527-567

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    Titre
    • On Kolmogorov equations for anisotropic multivariate Lévy processes
    Auteur
    • Reich, N.. Seminar for Applied Mathematics, ETH Zürich, 8092, Zürich, Switzerland
    • Schwab, C.. Seminar for Applied Mathematics, ETH Zürich, 8092, Zürich, Switzerland
    • Winter, C.. Seminar for Applied Mathematics, ETH Zürich, 8092, Zürich, Switzerland
    Type de document
    • Postprint
    Identifiant OAI-PMH
    • oai:doc.rero.ch:310067
    Summary
    For d-dimensional exponential Lévy models, variational formulations of the Kolmogorov equations arising in asset pricing are derived. Well-posedness of these equations is verified. Particular attention is paid to pure jump, d-variate Lévy processes built from parametric, copula dependence models in their jump structure. The domains of the associated Dirichlet forms are shown to be certain anisotropic Sobolev spaces. Singularity-free representations of the Dirichlet forms are given which remain bounded for piecewise polynomial, continuous functions of finite element type. We prove that the variational problem can be localized to a bounded domain with explicit localization error bounds. Furthermore, we collect several analytical tools for further numerical analysis