Approximation and simulation of infinite-dimensional Levy processes

TL;DR

This paper develops approximation and simulation methods for infinite-dimensional Levy fields, including a numerical Fourier inversion technique, with proven convergence properties and practical demonstrations on hyperbolic and Gaussian fields.

Contribution

It introduces a novel approach for simulating Levy fields with dependent components and provides convergence analysis for the approximation methods.

Findings

Effective numerical approximation of Levy measures using Fourier inversion.

Proven convergence rates for the approximation methods.

Numerical examples demonstrate the approach's efficiency on hyperbolic and Gaussian fields.

Abstract

In this paper approximation methods for infinite-dimensional Levy processes, also called (time-dependent) Levy 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-Loeve expansion a set of dependent but uncorrelated one-dimensional Levy 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) Levy-measures, a numerical method, called discrete Fourier inversion, is developed. For this method, $L^p$-convergence rates can be obtained and, under certain regularity assumptions, mean square and $L^p$-convergence of the approximated field is proved. Further, a class of (time-dependent) Levy 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.