Abstract
This paper constructs a Sparre Andersen risk model with a constant dividend barrier in which the claim interarrival distribution is a mixture of an exponential distribution and an Erlang(n) distribution. We derive the integro-differential equation satisfied by the Gerber-Shiu discounted penalty function of this risk model. Finally, we provide a numerical example.
Highlights
In recent years the Sparre Andersen model has been studied extensively
I=1 where u ≥ 0 represents the initial capital, c is the insurer’s rate of premium income per unit time, and {N(t), t ≥ 0} is the claim number process representing the number of claims up to time t. {Xi, i ≥ 1} is a sequence of i.i.d. random variables representing the individual claim amounts with distribution function F(x) and density function f(x) with mean μ
We study the Sparre Andersen risk model with a constant dividend barrier and the claim interarrival distribution is a mixture of an exponential distribution and an Erlang(n) distribution
Summary
We will study the time of ruin Tb and its related functions such as the surplus before ruin Ub(Tb−) and the deficit at ruin |Ub(Tb)|. If we let ω(x, y) = 1, (4) is the Laplace transform of the time of ruin Tb. If we let δ = 0 and ω(x, y) = 1, mb(u) becomes the ruin probability ψ(u). If we let δ = 0 and ω(x, y) = I(x ≤ z1)I(y ≤ z2), (4) becomes the joint df of the surplus before ruin and the deficit at ruin. If δ = 0 and ω(x, y) = x1n, we obtain the nth moment of the surplus before ruin. If δ = 0 and ω(x, y) = x2n, we obtain the nth moment of the deficit at ruin.
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