Sobolev spaces H m(x) (I) of variable order 0 < m(x) < 1 on an interval I ⊂ R arise as domains of Dirichlet forms for certain quadratic, pure jump Feller processes X t ∈ R with unbounded, state-dependent intensity of small jumps. For spline wavelets with complementary 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 Gårding inequality in H m(x) (I) and timeanalyticity of the semigroup T t associated with the Feller process X t are established. As application, wavelet-based algorithms of log-linear complexity are obtained for the valuation of contingent claims on pure jump Feller-Lévy processes X t with statedependent jump intensity by numerical solution of the corresponding Kolmogoroff equations.
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