Wavelet solution of variable order pseudodifferential equations. (English) Zbl 1202.65013
The authors obtain wavelet-based algorithms of log-linear complexity for the valuation of contingent claims on pure Feller-Lévy processes \(X_t\) with state-dependent jump intensity by a numerical solution of the corresponding Kolmogorov equations. They introduce a class of pseudodifferential operators \(A\) of variable order and derive estimates for the Schwartz kernels of \(A_X.\) They define variable order Sobolev spaces \(H^{m(x) },0\leq m(x) <1,\) which are the domains of Dirichlet forms of \(A_X\). For spline wavelets with complementary boundary conditions, they use the bounds on the Schwartz kernels to establish the main results of the paper: multilevel norm equivalences in \(H^{m(x) }(I) \) and compression estimates for the moment matrices of \(A_X\) in the wavelet basis. Sufficient conditions on \(A\) to satisfy a Gårding inequality in \(H^{m(x) }(I) \) and time-analyticity of the semigroup \(T_t\) associated with the Feller process \(X_t\) are demonstrated.
Reviewer: Adrian Carabineanu (Bucureşti)
MSC:
| 65C50 | Other computational problems in probability (MSC2010) |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65T60 | Numerical methods for wavelets |
| 47D07 | Markov semigroups and applications to diffusion processes |
| 65Y20 | Complexity and performance of numerical algorithms |
| 60G51 | Processes with independent increments; Lévy processes |
| 35S15 | Boundary value problems for PDEs with pseudodifferential operators |
| 91G60 | Numerical methods (including Monte Carlo methods) |
Keywords:
Feller processes; wavelets; pseudodifferential operators; analytic semigroups; option pricing; log-linear complexity; Feller-Lévy processes; Kolmogoroff equations; Schwartz kernels; Sobolev spaces; Dirichlet forms; multilevel norm equivalences; Gårding inequalityReferences:
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