Abstract
In this paper approximation methods for infinite-dimensional Lévy processes, also called (time-dependent) Lévy fields, are introduced. For square integrable fields beyond the Gaussian case, it is no longer given that the one-dimensional distributions in the spectral representation with respect to the covariance operator are independent. When simulated via a Karhunen–Loève expansion a set of dependent but uncorrelated one-dimensional Lévy processes has to be generated. The dependence structure among the one-dimensional processes ensures that the resulting field exhibits the correct point-wise marginal distributions. To approximate the respective (one-dimensional) Lévy-measures, a numerical method, called discrete Fourier inversion, is developed. For this method, Lp-convergence rates can be obtained and, under certain regularity assumptions, mean square and Lp-convergence of the approximated field is proved. Further, a class of (time-dependent) Lévy fields is introduced, where the point-wise marginal distributions are dependent but uncorrelated subordinated Wiener processes. For this specific class one may derive point-wise marginal distributions in closed form. Numerical examples, which include hyperbolic and normal-inverse Gaussian fields, demonstrate the efficiency of the approach.
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