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Simulation of infinitely divisible random fields

Wolfgang Karcher
(Ulm University, Germany)
Thiele Seminar
Thursday, 25 February, 2010, at 14:15-15:00, in Koll. D (1531-211)

Joint work with Hans-Peter Scheffler and Evgeny Spodarev

Let $\Lambda$ be an infinitely divisible random measure with Lebesgue control measure. We consider random fields of the form

$X(t) = \int_{\mathbb{R}^d} f_t(x) \Lambda(dx), \quad t \in \mathbb{R}^q, \quad d,q \geq 1,$
where $f_t:\mathbb{R}^d \to \mathbb{R}$ is $\Lambda$-integrable for all $t \in \mathbb{R}^d$. Our goal is to simulate sample paths of $X$ for kernel functions $f_t$ that are bounded or Hölder-continuous. We propose two efficient methods that are both based on approximating the kernel function appropriately such that the integral representation of $X$ reduces to a finite sum. In a simulation study, we compare the two approaches and outline the advantages and disadvantages. Finally, the simulation methods are used to assess the risk of a portfolio of storm insurance contracts by simulating claims from a fitted stable random field.

[1] Biermé, H. and Scheffler, H.-P. (2008): Fourier series approximation of linear fractional stable motion, Journal of Fourier Analysis and Applications, 14, 180 -- 202.
[2] Karcher, W., Scheffler, H.-P. and Spodarev, E. (2009): Simulation of infinitely divisible random fields, Preprint.

Organised by: The T.N. Thiele Centre
Contact person: Eva Bjørn Vedel Jensen