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The compound Poisson process perturbed by a diffusion with a threshold dividend strategy. (English) Zbl 1224.91100

This paper deals with the surplus process of an insurance company which follows the compound Poisson process perturbed by a diffusion \(X(t)=u+ct+\sigma W(t)-\sum_{k=1}^{N(t)}Z_{k}\), where \(W(t)\) is a standard Brownian motion; \(N(t)\) is a Poisson process with intensity \(\lambda\); \(Z_{k}, k=1,2,\dots\) are nonnegative i.i.d. claim-size random variables with mean \(\mu\). The authors consider the threshold strategy which assumes that dividends are paid at the maximal admissible rate \(\alpha<c\) whenever the surplus is above the threshold level \(b\), and that no dividends are paid whenever the surplus is below \(b\). It is assumed that \(c>\alpha+\lambda\mu\). Let \(D(t)\) denote the aggregate dividends paid by time \(t\). Let \(\delta>0\) be the force of interest. Let us denote the present value of all dividends until ruin by \(D=\int_{0}^{T}e^{-\delta t}dD(t)\), where \(T\) is the time of ruin for \(X(t)-D(t)\), and let \(V(u)=E[D\, | X(0)=u]\). The authors derive integro-differential equations for \(V(u)\) and obtain explicit infinite series expressions for the solution to the equations. Explicit expressions are proposed for the Laplace transform of the time of ruin and the Laplace transform of the aggregate dividends until ruin.

MSC:

91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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