Asymptotically optimal dividend policy for regime-switching compound Poisson models
type: Item
instance of: scholarly article
title: Asymptotically optimal dividend policy for regime-switching compound Poisson models
zbMATH Open document ID: 1204.91061
author: Zhuo Jin
author: Hailiang Yang
author: G. Yin
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 2010-10-29T00:00:00Z
review text: A Markov modulated risk model with dividends is considered, where the dividends are restricted to absolutely continuous payments with a maximal rate. The maximal rate is chosen, such that it is smaller than the average premium. Letting the environmental Markov chain \(\{\alpha(t)\}\) run faster (\(\alpha^\epsilon(t) = \alpha(t/\epsilon)\)), the risk process converges to the averaged process with intensity \(\lambda = \sum \nu_i \lambda_i\), claim size distribution \(F(x) = \sum \nu_i F_i(x)\) and premium intensity \(c = \sum \nu_i c_i\). Here \(\{\nu_i\}\) is the stationary distribution of \(\{\alpha(t)\}\), and \(\lambda_i\), \(F_i\), respectively, are the claim intensity and the claim size distribution when \(\alpha(t)\) is in state \(i\). The value of a dividend strategy \(\{\pi_i^\epsilon(t)\}\) is \[ J^\epsilon(x,i,\pi^\epsilon) = \mathbb{E}\Bigl[\int_0^\tau e^{-r t} \sum_i I_{\alpha^\epsilon(t)=i} \pi_i(t)\;d t\Bigr]\;, \] where \(\tau\) is the time of ruin. The value function is \(v^\epsilon(x,i) = \sup_{\pi^\epsilon} J^\epsilon(x,i,\pi^\epsilon)\). A strategy \(\pi^{\epsilon *}\) is called optimal if \(J^\epsilon(x,i,\pi^{\epsilon *}) = v^\epsilon(x,i)\). Now the limits of \(J^\epsilon(x,i,\pi^\epsilon)\), \(v^\epsilon(x,i)\) and \(\pi^{\epsilon *}\) as \(\epsilon \to 0\) are considered. Since the problem for the compound Poisson model is well studied, the results can be used as an approximation to the Markov modulated model. Here it had been interesting to make a numerical example to see how well the approximation works. A first problem is the differentiability of the value function. This is no problem if the minimal premium rate is larger than the maximal dividend rate. However, if there are states where more dividends can be paid than premia are earned it is possible that the derivative of the value function has jumps (see also the example in [\textit{P. Azcue} and \textit{N. Muler}, Math. Finance 15, 261--308 (2005; Zbl 1136.91016)]). A second problem is…
reviewed by: Hanspeter Schmidli
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 91G50
Mathematics Subject Classification ID: 91B70
zbMATH DE Number: 5808631
zbMATH Keywords: compound Poisson model
zbMATH Keywords: dividend policy
zbMATH Keywords: exponential claims
zbMATH Keywords: regime switching
zbMATH Keywords: average model
MaRDI profile type: Publication
full work available at URL: s10255-010-0023-0
OpenAlex ID: W2061200011
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DOI: 10.1007/S10255-010-0023-0
description: scientific article; zbMATH DE number 5808631