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The distribution of the dividend payments in the compound Poisson risk model perturbed by diffusion. (English) Zbl 1143.91032

This author deals with the classical surplus process perturbed by diffusion \[ U(t)=u+ct-\sum_{i=1}^{N(t)}X_{i}+\sigma B(t), \] where \(N(t)\) is a Poisson process; \(X_{1},X_{2}, \dots,\) are positive i.i.d. random variables, independent of \(N(t)\); \(B(t)\) is a standard Wiener process, independent of the aggregate claims process; \(u\) is the initial surplus. A barrier strategy is considered by assuming that there is a horizontal barrier of the level \(b\geq u\) such that when surplus reaches the level \(b\), dividends are paid continuously such that the surplus stays at level \(b\) until it become less than \(b\). Let \(U_{b}(t)\) be the modified surplus process, and let \(\delta>0\) be the force of interest. Define \(D_{u,b}=\int_{0}^{T_{b}}e^{-\delta t}dD(t)\), \(0\leq u\leq b\), to be the present value of all dividends until time of ruin \(T_{b}\), where \(D(t)\) is the aggregate dividends paid by time \(t\). The author derives the integro-differential equations for \(V_{n}(u;b)=E[D_{u,b}^{n}]\) and for \(M(u,y;b)=E[e^{yD_{u,b}}]\) and solves them in terms of the expected discounted penalty functions.

MSC:

91B30 Risk theory, insurance (MSC2010)
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