Recursive approximating to the finite-time Gerber-Shiu function in Lévy risk models under periodic observation
The surplus of an insurance company is modelled as a spectrally negative Lévy process \(U\). The surplus is observed only periodically with constant inter-observation times. Ruin occurs if the process is negative at some observation time, \(\tau = \inf\{t_j: U_{t_j} < 0\}\). The goal is to determine the Gerber-Shiu function \[ \phi(u) = \mathbb{E}[e^{-\delta \tau} w(|U_\tau|) 1_{\tau \le T} \mid U_0 = u]\;. \] Here \(w\) is a function such that \(\phi(u)\) becomes finite. It is assumed that the Lévy process contains a Gaussian component, such that a density of the increment distribution exists. Defining \(\phi_j(u) = \mathbb{E}[e^{-\delta (\tau-t_j)} w(|U_\tau|) 1_{\tau \le T} \mid U_0 = u, \tau > t_j]\). Then \(\phi(u) = \phi_0(u)\) can be calculated recursively. Using a Riesz basis, a numerical approximation of \(\phi_j\) can be obtained. The numerical errors are analysed. The discrete observations are important in order that only a finite number of recursions is needed. One could, instead, consider a model in discrete time with independent and stationary increments such that the increment distribution is in \(L^2\). Then the method would work, too.
scientific article; zbMATH DE number 7389244
scientific article