We consider the valuation of derivative contracts on baskets of risky assets whose prices are Lévy-like Feller processes of tempered stable type. The dependence among the marginals' jump structure is parametrized by a Lévy copula. For marginals of regular, exponential Lévy type in the sense of Ref. 7 we show that the infinitesimal generator [Formula: see text] of the resulting Lévy copula process is a pseudo-differential operator whose principal symbol is a distribution of anisotropic homogeneity. We analyze the jump measure of the corresponding Lévy copula processes. We prove that the domains of their infinitesimal generators [Formula: see text] are certain anisotropic Sobolev spaces. In these spaces and for a large class of Lévy copula processes, we prove a Gårding inequality for [Formula: see text]. We design a wavelet-based dimension-independent tensor product discretization for the efficient numerical solution of the parabolic Kolmogorov equation [Formula: see text] arising in valuation of derivative contracts under possibly stopped Lévy copula processes. In the wavelet basis, diagonal preconditioning yields a bounded condition number of the resulting matrices.

Lévy copulas, Lévy processes, Integro-differential equations, Pseudo-differential operators, Dirichlet forms, Option pricing, 45K05, 60J75, 65M60, C02,

Abstract. For a class of anisotropic integrodifferential operators B arising as semigroup generators ofMarkov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations Bu = f on [0,1] n with possibly large n. Under certain conditions on B, the scheme is of essentially optimal and dimension independent complexity O(h −1 | log h| 2(n−1) ) without corrupting the convergence or smoothness requirements of the original sparse tensor finite element scheme. If the conditions on B are not satisfied, the complexity can be bounded by O(h −(1+ε) ), where ε 1 tends to zero with increasing number of the wavelets' vanishing moments. Here h denotes the width of the corresponding finite element mesh. The operators under consideration are assumed to be of non-negative (anisotropic) order and admit a non-standard kernel κ(·, ·) that can be singular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented.Mathematics Subject Classification. 47A20, 65F50, 65N12, 65Y20, 68Q25, 45K05, 65N30.

Multidimensional Lévy processes, Numerical methods, Asset pricing, 60J75, 65N06, 65N30, C63,

Abstract. It is shown that infinitesimal generators A of certain multivariate pure jump Lévy copula processes give rise to a class of anisotropic symbols that extends the well-known classes of pseudo differential operators of Hörmander-type. In addition, we provide minimal regularity convergence analysis for a sparse tensor product finite element approximation to solutions of the corresponding stationary Kolmogorov equations Au = f . The computational complexity of the presented approximation scheme is essentially independent of the underlying state space dimension. Mathematics Subject Classification (2000). Primary 45K05, 60J75, 47G30; Secondary 65N30, 47B38.