Recursive approximating to the finite-time Gerber-Shiu function in Lévy risk models under periodic observation
type: Item
instance of: scholarly article
title: Recursive approximating to the finite-time Gerber-Shiu function in Lévy risk models under periodic observation
zbMATH Open document ID: 1476.91038
author: Yanyan Li
published in: Journal of Computational and Applied Mathematics
publication date: 2021-09-01T00:00:00Z
review text: The surplus of an insurance company is modelled as a spectrally negative Lévy process \(U\). The surplus is observed only periodically with constant inter-observation times. Ruin occurs if the process is negative at some observation time, \(\tau = \inf\{t_j: U_{t_j} < 0\}\). The goal is to determine the Gerber-Shiu function \[ \phi(u) = \mathbb{E}[e^{-\delta \tau} w(|U_\tau|) 1_{\tau \le T} \mid U_0 = u]\;. \] Here \(w\) is a function such that \(\phi(u)\) becomes finite. It is assumed that the Lévy process contains a Gaussian component, such that a density of the increment distribution exists. Defining \(\phi_j(u) = \mathbb{E}[e^{-\delta (\tau-t_j)} w(|U_\tau|) 1_{\tau \le T} \mid U_0 = u, \tau > t_j]\). Then \(\phi(u) = \phi_0(u)\) can be calculated recursively. Using a Riesz basis, a numerical approximation of \(\phi_j\) can be obtained. The numerical errors are analysed. The discrete observations are important in order that only a finite number of recursions is needed. One could, instead, consider a model in discrete time with independent and stationary increments such that the increment distribution is in \(L^2\). Then the method would work, too.
reviewed by: Hanspeter Schmidli
Mathematics Subject Classification ID: 91B05
Mathematics Subject Classification ID: 65D15
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60K10
zbMATH DE Number: 7389244
zbMATH Keywords: Lévy risk model
zbMATH Keywords: finite-time Gerber-Shiu function
zbMATH Keywords: periodic observation
zbMATH Keywords: frame duality projection
MaRDI profile type: Publication
full work available at URL: j.cam.2021.113703
OpenAlex ID: W3178651805
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DOI: 10.1016/J.CAM.2021.113703
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description: scientific article; zbMATH DE number 7389244