Μέτρα χρεοκοπίας για ανανεωτικές στοχαστικές διαδικασίες πλεονάσματος με εξάρτηση

Σπαντίδος, Ευστράτιος

Ruin measures for stochastic surplus processes with dependence

Master Thesis

Author

Σπαντίδος, Ευστράτιος

Date

2025-02

Abstract

Chapter 1 introduces fundamental mathematical concepts and the classical Poisson risk model, commonly used in analyzing the solvency of insurance companies. It presents Lagrange polynomials, the Laplace transform, the Dickson-Hipp operator, and the deficit renewal equation, providing tools for understanding surplus through stochastic processes. Additionally, it examines mixed Erlang and Coxian distributions, with the compound Poisson distribution modeling total risk. Finally, it discusses ruin measures and the Gerber-Shiu function, analyzing surplus and deficit values during ruin scenarios. Chapter 2 focuses on the generalization of the classical Poisson risk model, introducing the dependent Sparre-Andersen model. This model enables the analysis of the distributions of claims and surplus, considering dependencies among random variables. The Gerber-Shiu function is analyzed, examining bankruptcy, deficit, and surplus, with equations provided to describe these phenomena, including Laplace transformations. Lastly, the moments of the deficit at the time of bankruptcy are explored, revealing critical characteristics of the model. Chapter 3 introduces a dependent Sparre Andersen model where the time between claims and their sizes follow a Coxian distribution and may be interdependent. It examines the moments of the time to ruin using systems of linear equations and explores the Gerber-Shiu function to describe surplus behavior. The chapter presents theorems offering analytical solutions and methodologies for calculating these moments. Lastly, it provides a numerical example for two dependency cases, along with R code and graphical representations of the expected time and variance of ruin as functions of initial surplus. In chapter 4 explores the extension of the Sparre Andersen model using the Farlie–Gumbel–Morgenstern (FGM) copula to describe dependencies between claim sizes and inter-arrival times. Arrival times are modeled using the Erlang distribution. Additionally, the Gerber–Shiu function is examined, providing expressions for ruin probabilities and the generalized Lundberg equation. Detailed results for exponentially distributed claims are also presented. Finaly, Chapter 5 explores the dependency between random variables using the Spearman copula, extending the classical risk model by removing the assumption of independence between claims and inter-arrival times. It analyzes the generalized Lundberg equation, the Laplace transform of the penalty function, and the probability of ruin. The chapter examines the impact of dependency degrees on ruin probabilities and provides numerical examples for the Gerber-Shiu functions and ruin probabilities, considering various dependency parameter values.

Postgraduate Studies Programme

Αναλογιστική Επιστήμη και Διαχείριση Κινδύνων

Department

Σχολή Χρηματοοικονομικής και Στατιστικής. Τμήμα Στατιστικής και Ασφαλιστικής Επιστήμης

Number of pages

101

Language

Greek

URI

https://dione.lib.unipi.gr/xmlui/handle/unipi/17784

Collections

Τμήμα Στατιστικής και Ασφαλιστικής Επιστήμης

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Αναφορά Δημιουργού-Μη Εμπορική Χρήση-Όχι Παράγωγα Έργα 3.0 Ελλάδα