On the Discounted Penalty Function in a Perturbed Erlang Renewal Risk Model With Dependence

In this paper, we consider the risk model perturbed by a diffusion process. We assume an Erlang(n) risk process, ( n = 1 , 2 , … ) to study the Gerber-Shiu discounted penalty function when ruin is due to claims or oscillations by including a dependence structure between claim sizes and their occurre...

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Bibliographic Details
Published in:Methodology and computing in applied probability 2022-06, Vol.24 (2), p.481-513
Main Authors: Adékambi, Franck, Takouda, Essodina
Format: Article
Language:English
Subjects:
Business and Management
Differential equations
Economics
Electrical Engineering
Laplace transforms
Life Sciences
Mathematics and Statistics
Penalty function
Probability
Risk
Statistics
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Summary:In this paper, we consider the risk model perturbed by a diffusion process. We assume an Erlang(n) risk process, ( n = 1 , 2 , … ) to study the Gerber-Shiu discounted penalty function when ruin is due to claims or oscillations by including a dependence structure between claim sizes and their occurrence time. We derive the integro-differential equation of the expected discounted penalty function, its Laplace transform. Then, by analyzing the roots of the generalized Lundberg equation, we show that the expected penalty function satisfies a certain defective renewal equation and provide its representation solution. Finally, we give some explicit expressions for the Gerber-Shiu discounted penalty functions when the claim size distributions are Erlang(m), ( m = 1 , 2 , … ) and provide numerical examples to illustrate the ruin probability.
ISSN:1387-5841
1573-7713
DOI:10.1007/s11009-022-09944-3