A class of Sparre Andersen risk process (Q610720)

A Sparre Andersen renewal risk model \(U(t)\) is considered, where the interarrival times are mixed Erlang distributed, with density \[ g(x) = p {\lambda^m x^{m-1} \over (m-1)! } e^{-\lambda x} + (1-p) {\lambda^n x^{n-1} \over (n-1)! } e^{-\lambda x} \;. \] Here \(p \in (0,1)\), \(\lambda > 0\) and \(n \geq m \geq 1\). The claim size distribution is assumed to be absolutely continuous with density \(f(y)\). The quantity of interest is the Gerber-Shiu function \[ \Phi(u) = \mathbb{E} [e^{-\delta T} w(U(T-),-U(T)) 1_{T < \infty} \mid U(0) = u]\;, \] where \(\delta \geq 0\), \(T = \inf\{t> 0; U(t) < 0\}\) is the time of ruin, and \(w(x,y)\) is a non-negative bounded function. Repeating the approach of \textit{H. U. Gerber} and \textit{E. S.W. Shiu} [N. Am. Actuar. J. 9, No. 2, 49--84 (2005; Zbl 1085.62508)], the Laplace transform \(\hat\Phi(s) = \int_0^\infty e^{-s u} \Phi(u)\;\text{d} u\) of \(\Phi\) is found. As commented in the discussion by Schmidli, the approach of Gerber and Shiu easily can be extended to phase-type distributions. The function \(g(x)\) is just a special case of a phase-type distribution. From the Laplace transform, a defective renewal equation can be found by the method of \textit{Shuanming Li} (discussion of \textit{Y. Cheng} and \textit{Q. Tang} [N. Am. Actuar. J. 7, No. 1, 1--12 (2003; Zbl 1084.60544)]). Using renewal theory, the asymptotic behaviour of the Gerber-Shiu function in the small claim case is found as \(u\) tends to infinity. Finally, the asymptotic behaviour in the subexponential case is obtained. In the small claim case it is assumed that \(\int_u^\infty w(u,y-u) f(y)\; \text{d} y e^{R u}\) tends to zero. As, for example, shown in [\textit{H. Schmidli}, Insur. Math. Econ. 46, No. 1, 3--11 (2010)], it is enough to assume that \(\int_u^\infty w(u,y-u) e^{R y} f(y)\; \text{d} y\) is directly Riemann integrable. The result for subexponential claim sizes only holds in the case where \(\delta = 0\), since in the proof it is assumed that \(0\) is a root of Lundberg's fundamental equation. In this case, the asymptotic behaviour of the distribution prior and at ruin is discussed extensively in the literature. Therefore, also the asymptotic behaviour of the Gerber-Shiu function is known.