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

Franck Adékambi () and Essodina Takouda ()
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Franck Adékambi: University of Johannesburg
Essodina Takouda: University of Johannesburg

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 2, 481-513

Abstract: Abstract In this paper, we consider the risk model perturbed by a diffusion process. We assume an Erlang(n) risk process, ( $$n=1,2,\ldots$$ 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), ( $$m=1,2,\ldots$$ m = 1 , 2 , … ) and provide numerical examples to illustrate the ruin probability.

Keywords: Ruin theory; Aggregate risk process; Renewal equation; Convolution formula; Diffusion process; Erlang distribution; Copulas; Penalty function (search for similar items in EconPapers)
Date: 2022
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s11009-022-09944-3

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