Wavelet solution of variable order pseudodifferential equations

Sobolev spaces Hm(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 Xt∈ℝ with unbounded, state-dependent intensity of small jumps. For spline wavelets with complementary boundary conditions, we establish multilevel norm equivalences in Hm(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 Hm(x)(I) and time-analyticity of the semigroup Tt associated with the Feller process Xt 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 Xt with state-dependent jump intensity by numerical solution of the corresponding Kolmogoroff equations.

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