Savo darbe mes nagrinėjame Gerber-Shiu baudos funkciją klasikiniame rizikos modelyje atveju, kai žalų dydžiai pasiskirstę pagal Pareto dėsnį. Pagrindinis uždavinys yra susikonstruoti algoritmą funkcijos reikšmių gavimui. Tiriamas Gerber-Shiu diskontuotos baudos funkcijos atvejis, kada vidinė baudos funkcija w tapačiai lygi vienetui. Dėl sudėtingos transformuoto Pareto skirstinio formos analitiškai paskaičiuoti sąsūkų nepavyko. Tam tikslui naudojamas interpoliavimas kubiniu splainu. N kartų kartodami sukonstruotą algoritmą gauname pirmąsias n sąsūkas laisvai pasirinktiems pradiniams parametrams: Pareto skirstinio laipsnio rodikliui α, pradiniam kapitalui u, santykinei draudimo priemokai θ, diskontavimo parametrui (palūkanų normai) δ ir Puasono proceso parametrui λ. Lentelių pagalba parodome funkcijos priklausomybę nuo skirtingų modeliuojančių parametrų reikšmių. Išvadose teigiame jog pasiūlytas metodas skaičiuoti Gerber-Shiu diskontuotos baudos funkciją nors ir išpildomas tačiau yra neefektyvus. Kai kuriais pradinių parametrų pasirinkimo atvejais susiduriama su tikslumo problema. Norint tiksliai paskaičiuoti funkcijos reikšmes reikia didesnių eilių transformuoto Pareto skirstinio sąsūkų, o tam reikalingi dideli resursai. Kita vertus, pradinio kapitalo u reikšmėms didėjant tikslumas didėja ženkliai. / In this paper we consider Gerber-Shiu discounted penalty function in the classical risk model for Pareto claims. Our main goal is to construct an algorithm for obtaining values of the discounted penalty function (considering penalty function w=1). Due to the complicated form of the transformed Pareto distribution function we cannot obtain its convolutions analiticaly. We use numerical methods provided by Maple (cube spline) to find interpolating functions instead. Continuously applying recursive formulas we obtain first 5 interpolated convolutions. Then we calculate values of Gerber-Shiu discounted penalty function for certain arbitrary parameters: α – degree of Pareto distribution function, initial surplus u, security loading θ, discounting parameter δ and Poison process parameter λ. We present data tables and graphs of the discounted penalty function for some variations of parameters in later sections. Finally we state that the method that we use is quite complicated. For better accuracy of the discounted penalty function values one may require to get many convolutions of the transformed Pareto distribution function and that may require too great of the resources. However the quantity of the convolutions needed rapidly decreases for large values of the initial surplus u.

Darbe gauta Gerber-Shiu diskontuotos baudos funkcijos asimptotika, kai žalos pasiskirsčiusios pagal Pareto dėsnį ir pradinis kapitalas x artėja į begalybę. Pagrindinė išraiška Gerber-Shiu diskontuotos baudos funkcijos išskaidyta į du atvejus, kai palūkanų norma nelygi ir lygi nuliui. Darbe pateikti grafikai rodo diskontuotos baudos funkcijos priklausomybę nuo įvairių Puasono modelio parametrų. / The asymptotic of the Gerber-Shiu discounted penalty function in Poisson model with Pareto distributed claims is obtained. The asymptotic is obtained as initial surplus x tends to infinity. The main term of discounted penalty function has different expression in case when interest rate equal zero and when doesn't equal zero. The graphs of the Gerber-Shiu discounted penalty function in the case of Pareto claims are examined for various parameter choices.

Darbe apskaičiuotas Gerber-Shiu diskontuotos baudos funkcijos pagrindinis narys klasikiniame kolektyvinės rizikos modelyje, kai draudimo kompanijos žalos pasiskirsčiusios pagal Veibulo skirstinį su parametrais &#951; = const, 0< &#951; <1 ir &#963; = 1, o pradinis kompanijos turtas . Minėtojo nario asimptotika gauta pasinaudojus subeksponentinių pasiskirstymo funkcijų savybėmis. Darbe pateiktuose grafikuose pavaizduota diskontuotos baudos funkcijos pagrindinio nario priklausomybė nuo įvairių klasikinio kolektyvinės rizikos modelio parametrų. / In this work the main member of the Gerber-Shiu discounted penalty function in a classic collective risk model with Weibull distribution (parameters &#951; = const, 0< &#951; <1 and &#963; = 1) is calculated. The expression of the main member is obtained by making use of properties of subexponential distribution functions. In the graphs a dependence of the main member of the Gerber-Shiu discounted penalty function on various parameters of classic collective risk model is represented.

On présente une nouvelle approche de simulation pour la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine, pour des modèles de risque déterminés par des subordinateurs de Lévy. Cette approche s'inspire de la décomposition "Ladder height" pour la probabilité de ruine dans le Modèle Classique. Ce modèle, déterminé par un processus de Poisson composé, est un cas particulier du modèle plus général déterminé par un subordinateur, pour lequel la décomposition "Ladder height" de la probabilité de ruine s'applique aussi. La Fonction de Pénalité Escomptée, encore appelée Fonction Gerber-Shiu (Fonction GS), a apporté une approche unificatrice dans l'étude des quantités liées à l'événement de la ruine été introduite. La probabilité de ruine et la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine sont des cas particuliers de la Fonction GS. On retrouve, dans la littérature, des expressions pour exprimer ces deux quantités, mais elles sont difficilement exploitables de par leurs formes de séries infinies de convolutions sans formes analytiques fermées. Cependant, puisqu'elles sont dérivées de la Fonction GS, les expressions pour les deux quantités partagent une certaine ressemblance qui nous permet de nous inspirer de la décomposition "Ladder height" de la probabilité de ruine pour dériver une approche de simulation pour cette fonction de densité conjointe. On présente une introduction détaillée des modèles de risque que nous étudions dans ce mémoire et pour lesquels il est possible de réaliser la simulation. Afin de motiver ce travail, on introduit brièvement le vaste domaine des mesures de risque, afin d'en calculer quelques unes pour ces modèles de risque. Ce travail contribue à une meilleure compréhension du comportement des modèles de risques déterminés par des subordinateurs face à l'éventualité de la ruine, puisqu'il apporte un point de vue numérique absent de la littérature. / We discuss a simulation approach for the joint density function of the surplus prior to ruin and deficit at ruin for risk models driven by Lévy subordinators. This approach is inspired by the Ladder Height decomposition for the probability of ruin of such models. The Classical Risk Model driven by a Compound Poisson process is a particular case of this more generalized one. The Expected Discounted Penalty Function, also referred to as the Gerber-Shiu Function (GS Function), was introduced as a unifying approach to deal with different quantities related to the event of ruin. The probability of ruin and the joint density function of surplus prior to ruin and deficit at ruin are particular cases of this function. Expressions for those two quantities have been derived from the GS Function, but those are not easily evaluated nor handled as they are infinite series of convolutions with no analytical closed form. However they share a similar structure, thus allowing to use the Ladder Height decomposition of the Probability of Ruin as a guiding method to generate simulated values for this joint density function. We present an introduction to risk models driven by subordinators, and describe those models for which it is possible to process the simulation. To motivate this work, we also present an application for this distribution, in order to calculate different risk measures for those risk models. An brief introduction to the vast field of Risk Measures is conducted where we present selected measures calculated in this empirical study. This work contributes to better understanding the behavior of subordinators driven risk models, as it offers a numerical point of view, which is absent in the literature.

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas.

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas.

On présente une nouvelle approche de simulation pour la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine, pour des modèles de risque déterminés par des subordinateurs de Lévy. Cette approche s'inspire de la décomposition "Ladder height" pour la probabilité de ruine dans le Modèle Classique. Ce modèle, déterminé par un processus de Poisson composé, est un cas particulier du modèle plus général déterminé par un subordinateur, pour lequel la décomposition "Ladder height" de la probabilité de ruine s'applique aussi. La Fonction de Pénalité Escomptée, encore appelée Fonction Gerber-Shiu (Fonction GS), a apporté une approche unificatrice dans l'étude des quantités liées à l'événement de la ruine été introduite. La probabilité de ruine et la fonction de densité conjointe du surplus avant la ruine et du déficit au moment de la ruine sont des cas particuliers de la Fonction GS. On retrouve, dans la littérature, des expressions pour exprimer ces deux quantités, mais elles sont difficilement exploitables de par leurs formes de séries infinies de convolutions sans formes analytiques fermées. Cependant, puisqu'elles sont dérivées de la Fonction GS, les expressions pour les deux quantités partagent une certaine ressemblance qui nous permet de nous inspirer de la décomposition "Ladder height" de la probabilité de ruine pour dériver une approche de simulation pour cette fonction de densité conjointe. On présente une introduction détaillée des modèles de risque que nous étudions dans ce mémoire et pour lesquels il est possible de réaliser la simulation. Afin de motiver ce travail, on introduit brièvement le vaste domaine des mesures de risque, afin d'en calculer quelques unes pour ces modèles de risque. Ce travail contribue à une meilleure compréhension du comportement des modèles de risques déterminés par des subordinateurs face à l'éventualité de la ruine, puisqu'il apporte un point de vue numérique absent de la littérature. / We discuss a simulation approach for the joint density function of the surplus prior to ruin and deficit at ruin for risk models driven by Lévy subordinators. This approach is inspired by the Ladder Height decomposition for the probability of ruin of such models. The Classical Risk Model driven by a Compound Poisson process is a particular case of this more generalized one. The Expected Discounted Penalty Function, also referred to as the Gerber-Shiu Function (GS Function), was introduced as a unifying approach to deal with different quantities related to the event of ruin. The probability of ruin and the joint density function of surplus prior to ruin and deficit at ruin are particular cases of this function. Expressions for those two quantities have been derived from the GS Function, but those are not easily evaluated nor handled as they are infinite series of convolutions with no analytical closed form. However they share a similar structure, thus allowing to use the Ladder Height decomposition of the Probability of Ruin as a guiding method to generate simulated values for this joint density function. We present an introduction to risk models driven by subordinators, and describe those models for which it is possible to process the simulation. To motivate this work, we also present an application for this distribution, in order to calculate different risk measures for those risk models. An brief introduction to the vast field of Risk Measures is conducted where we present selected measures calculated in this empirical study. This work contributes to better understanding the behavior of subordinators driven risk models, as it offers a numerical point of view, which is absent in the literature.