Η συνάρτηση των Gerber - Shiu. Μελέτη του χρόνου χρεοκοπίας και μέτρων κινδύνου σε στοχαστικές διαδικασίες πλεονάσματος
The Gerber - Shiu function. Study of ruin time and risk measures for stochastic surplus process
Κολιού, Ηλιάνα Δ.
KeywordsΘεωρία χρεοκοπίας ; Διαχείριση κινδύνου ; Στοχαστικές διαδικασίες ; Ανανεωτικές εξισώσεις ; Μετασχηματισμοί Laplace ; Συνάρτηση Gerber-Shiu
In risk theory, the time to ruin is one of the central quantities. The Laplace transform, density and moments of the time to ruin have been studied by many authors under different risk model assumptions. The Gerber-Shiu function proposed by Gerber and Shiu (1998) provides an analytic tool in studying these quantities. For example, Dickson and Willmot (2005) inverted the Gerber-Shiu function with respect to the Laplace transform parameter of the time to ruin by Lagrange's implicit function theorem, and hence obtained the density of the time to ruin. The main focus of this thesis is to study the moments involving the time to ruin by using Gerber-Shiu function as the analytic tool. An introduction on the Gerber-Shiu function and different risk models is first given in Chapter 1. In Chapter 2, the moments of the time to ruin are studied as generalized versions of the Gerber-Shiu function in dependent Sparre Andersen models. It is shown that structural properties of the Gerber-Shiu function hold also for the moments of the time to ruin. In particular, the moments continue to satisfy defective renewal equations. These properties are discussed in detail in Chapter 4 under the model of Willmot and Woo (2012) where Coxian interclaim times and arbitrary time-dependent claim sizes are assumed. In Chapter 3, another very general class of dependent Sparre Andersen models with Coxian claim sizes (e.g. Landriault et al. (2014)) is considered. An analytical form is provided for the moments of the time to ruin under this class, which involves solving linear systems of equations. In Chapter 5, the number of claims until ruin is taken into consideration under a Sparre Andersen model with exponential claim sizes. The joint density of the time to ruin, the number of claims until ruin and other ruin-related quantities is identified. The joint moments of these quantities can then be obtained from this joint density.