Wavelet solution of variable order pseudodifferential equations (Q987714)

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.