Anisotropic Operator Symbols Arising From Multivariate Jump Processes

It is shown that infinitesimal generators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}}$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{A}}u = f$$\end{document}. The computational complexity of the presented approximation scheme is essentially independent of the underlying state space dimension.