Gerber-Shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times
type: Item
instance of: scholarly article
title: Gerber-Shiu analysis in a perturbed risk model with dependence between claim sizes and interclaim times
zbMATH Open document ID: 1202.91131
author: Zhimin Zhang
author: Hu Yang
published in: Journal of Computational and Applied Mathematics
publication date: 2010-11-30T00:00:00Z
review text: The authors analyse a compound Poisson risk model perturbed by a Brownian motion, that is \[ U(t)=u+ct-S(t)+\sigma B(t), \] where \(u\geq0\) is the initial surplus, \(c>0\) the premium rate, \(S(t)\) the aggregate claim process, \(B(t)\) a standard Brownian motion starting from \(0\) and \(\sigma\) is the diffusion volatility. In particular \[ S(t)=\sum_{i=1}^{N(t)}X_i, \] \(\{N(t), t\geq0\}\) being a Poisson process which depicts the number of claims up to \(t\) and \(\{X_i, i\geq 1\}\) being a sequence of strictly positive random variables which represent the individual claim sizes. A central role is played by the sequence of random variables \(\{V_i, i\geq 1\}\), representing the interclaim times. In fact the paper focuses on the case of a perturbed risk model with dependence, i.e. under the assumption that the claim size and the interclaim time have a certain bivariate c.d.f. Within this context, modelling the c.d.f. by the Farlie-Gumbel-Morgenstern copula, the integro-differential equations involving the Gerber-Shiu functions are obtained. Some examples are given in the case of exponential claims.
reviewed by: Emilia Di Lorenzo
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 91B70
Mathematics Subject Classification ID: 60K20
zbMATH DE Number: 5821392
zbMATH Keywords: Gerber-Shiu function
zbMATH Keywords: integro-differential equation
zbMATH Keywords: Laplace transform
zbMATH Keywords: ruin probability
MaRDI profile type: Publication
full work available at URL: j.cam.2010.08.003
OpenAlex ID: W2103676881
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DOI: 10.1016/J.CAM.2010.08.003
description: scientific article; zbMATH DE number 5821392