Reinhold Schneider, Christoph Schwab:

Wavelet FEM for variable order pseudodifferential equations

Sobolev spaces H^{m(x)}(I) of variable order 0 < m(x) < 1 on an interval I\subset \IR arise as domains of Dirichlet forms for certain quadratic, pure jump Feller processes X_t\in\mathbb{R} with unbounded, state-dependent intensity of small jumps. For spline wavelets with complementing boundary conditions, we establish multilevel norm equivalences in H^{m(x)}(I) and prove preconditioning and wavelet matrix compression results for the variable order pseudodifferential generators A of X.

Sufficient conditions on A to satisfy a Garding inequality in H^{m(x)}(I) and time-analyticity of the semigroup T_t associated with the Feller process X_t are established.

As application, wavelet-based algorithms with discontinuous Galerkin time-stepping of log-linear complexity for the valuation of contingent claims on pure jump Feller-Lévy processes X_t with state-depenent jump intensity by numerical solution of the corresponding Kolmogoroff equations which are parabolic pseudodifferential equations of variable order are obtained.

Mathematics Subject Classification (1991): 35K15, 45K05, 65N30

Keywords: Feller processes, Wavelets, Pseudodifferential Operators, Analytic Semigroups, Option Pricing

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[Thu Feb 19 18:56:36 2009]

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