We study the infinite time ruin probability for an insurance company in the classical Cramér-Lundberg model with finite exponential moments. The additional non-classical feature is that the company is also allowed to invest in some stock market, modeled by geometric Brownian motion. We obtain an exact analogue of the classical estimate for the ruin probability without investment, i.e., an exponential inequality. The exponent is larger than the one obtained without investment, the classical Lundberg adjustment coefficient, and thus one gets a sharper bound on the ruin probability. A surprising result is that the trading strategy yielding the optimal asymptotic decay of the ruin probability simply consists in holding a fixed quantity (which can be explicitly calculated) in the risky asset, independent of the current reserve. This result is in apparent contradiction to the common believe that 'rich' companies should invest more in risky assets than 'poor' ones. The reason for this seemingly paradoxical result is that the minimization of the ruin probability is an extremely conservative optimization criterion, especially for 'rich' companies. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

Apresenta-se uma metodologia de projeto de fundações por estacas incluindo probabilidade de ruína. Considera-se que a complexidade do comportamento geológico-geotécnico do maciço de solo e do elemento estrutural de fundação, submetidos às ações aleatórias ambientais e funcionais, pode ser avaliada por meio das variáveis aleatórias resistência e solicitação. Estatisticamente, a metodologia proposta supõe que a população da fundação é finita, e que os estimadores extraídos das sondagens de simples reconhecimento e das provas de carga podem ser representativos da população (Aoki, 2002), também podem ser avaliados pelas estatísticas de ordem. Aplica-se esta metodologia na fundação do Píer 3 do Porto de Vila do Conde localizado no Estado do Pará, tendo como compartimentação geológica a formação Barreiras. Conclui-se, de modo geral, que esta metodologia pode ser aplicada nas obras de fundações por estacas, especialmente, para auxiliar nas tomadas de decisões. / A methodology of pile foundation projects is presented incorporating ruin probability. It is considered that the complexity of the geological and geotechnical behaviors of the soil mass and the structural element of foundation, subjected to environmental and functional random actions, can be evaluated by strength and solicitation variables. Statistically, the proposed methodology assumes that the population of the foundation is finite, and that the extracted estimators of the standard penetration tests (SPT) and the load tests could be representative of the population (Aoki, 2002). The extracted estimators of the said tests can also be evaluated by the order statistics. This methodology is applied in the Pier 3 foundation of Porto de Vila do Conde, Pará State, which is underlain by the Barreiras formation. It is concluded that in most cases, this methodology could be applied to pile foundations, especially, in taking decisions.

In this thesis we study the asymptotic behaviour of the solution of a discrete time renewal equation depending on a small perturbation parameter. In particular, we construct asymptotic expansions for the solution of the renewal equation and related quantities. The results are applied to studies of quasi-stationary phenomena for regenerative processes and asymptotics of ruin probabilities for a discrete time analogue of the Cramér-Lundberg risk model.

Apresenta-se uma metodologia de projeto de fundações por estacas incluindo probabilidade de ruína. Considera-se que a complexidade do comportamento geológico-geotécnico do maciço de solo e do elemento estrutural de fundação, submetidos às ações aleatórias ambientais e funcionais, pode ser avaliada por meio das variáveis aleatórias resistência e solicitação. Estatisticamente, a metodologia proposta supõe que a população da fundação é finita, e que os estimadores extraídos das sondagens de simples reconhecimento e das provas de carga podem ser representativos da população (Aoki, 2002), também podem ser avaliados pelas estatísticas de ordem. Aplica-se esta metodologia na fundação do Píer 3 do Porto de Vila do Conde localizado no Estado do Pará, tendo como compartimentação geológica a formação Barreiras. Conclui-se, de modo geral, que esta metodologia pode ser aplicada nas obras de fundações por estacas, especialmente, para auxiliar nas tomadas de decisões. / A methodology of pile foundation projects is presented incorporating ruin probability. It is considered that the complexity of the geological and geotechnical behaviors of the soil mass and the structural element of foundation, subjected to environmental and functional random actions, can be evaluated by strength and solicitation variables. Statistically, the proposed methodology assumes that the population of the foundation is finite, and that the extracted estimators of the standard penetration tests (SPT) and the load tests could be representative of the population (Aoki, 2002). The extracted estimators of the said tests can also be evaluated by the order statistics. This methodology is applied in the Pier 3 foundation of Porto de Vila do Conde, Pará State, which is underlain by the Barreiras formation. It is concluded that in most cases, this methodology could be applied to pile foundations, especially, in taking decisions.

<p>This thesis deals with a model of nonlinearly perturbed continuous-time renewal equation with nonpolynomial perturbations. The characteristics, namely the defect and moments, of the distribution function generating the renewal equation are assumed to have expansions with respect to a non-polynomial asymptotic scale: $\{\varphi_{\nn} (\varepsilon) =\varepsilon^{\nn \cdot \w}, \nn \in \mathbf{N}_0^k\}$  as $\varepsilon \to 0$, where $\mathbf{N}_0$ is the set of non-negative integers, $\mathbf{N}_0^k \equiv \mathbf{N}_0 \times \cdots \times \mathbf{N}_0, 1\leq k <\infty$ with the product being taken $k$ times and $\w$ is a $k$ dimensional parameter vector that satisfies certain properties. For the one-dimensional case, i.e., $k=1$, this model reduces to the model of nonlinearly perturbed renewal equation with polynomial perturbations which is well studied in the literature.  The goal of the present study is to obtain the exponential asymptotics for the solution to the perturbed renewal equation in the form of exponential asymptotic expansions and present possible applications.</p><p>The thesis is based on three papers which study successively the model stated above. Paper A investigates the two-dimensional case, i.e. where $k=2$. The corresponding asymptotic exponential expansion for the solution to the perturbed renewal equation is given. The asymptotic results are applied to an example of the perturbed risk process, which leads to diffusion approximation type asymptotics for the ruin probability.  Numerical experimental studies on this example of perturbed risk process are conducted in paper B, where Monte Carlo simulation are used to study the accuracy and properties of the asymptotic formulas. Paper C presents the asymptotic results for the more general case where the dimension $k$ satisfies $1\leq k <\infty$, which are applied to the asymptotic analysis of the ruin probability in an example of perturbed risk processes with this general type of non-polynomial perturbations.  All the proofs of the theorems stated in paper C are collected in its supplement: paper D.</p>

This thesis deals with a model of nonlinearly perturbed continuous-time renewal equation with nonpolynomial perturbations. The characteristics, namely the defect and moments, of the distribution function generating the renewal equation are assumed to have expansions with respect to a non-polynomial asymptotic scale: $\{\varphi_{\nn} (\varepsilon) =\varepsilon^{\nn \cdot \w}, \nn \in \mathbf{N}_0^k\}$  as $\varepsilon \to 0$, where $\mathbf{N}_0$ is the set of non-negative integers, $\mathbf{N}_0^k \equiv \mathbf{N}_0 \times \cdots \times \mathbf{N}_0, 1\leq k &lt;\infty$ with the product being taken $k$ times and $\w$ is a $k$ dimensional parameter vector that satisfies certain properties. For the one-dimensional case, i.e., $k=1$, this model reduces to the model of nonlinearly perturbed renewal equation with polynomial perturbations which is well studied in the literature.  The goal of the present study is to obtain the exponential asymptotics for the solution to the perturbed renewal equation in the form of exponential asymptotic expansions and present possible applications. The thesis is based on three papers which study successively the model stated above. Paper A investigates the two-dimensional case, i.e. where $k=2$. The corresponding asymptotic exponential expansion for the solution to the perturbed renewal equation is given. The asymptotic results are applied to an example of the perturbed risk process, which leads to diffusion approximation type asymptotics for the ruin probability.  Numerical experimental studies on this example of perturbed risk process are conducted in paper B, where Monte Carlo simulation are used to study the accuracy and properties of the asymptotic formulas. Paper C presents the asymptotic results for the more general case where the dimension $k$ satisfies $1\leq k &lt;\infty$, which are applied to the asymptotic analysis of the ruin probability in an example of perturbed risk processes with this general type of non-polynomial perturbations.  All the proofs of the theorems stated in paper C are collected in its supplement: paper D.

During the last decade several Japanese life insurance companies failed mainly due to interest losses. In fact, interest rate risk dominates mortality risk for a portfolio of business in force. When the interest rates are modelled as random variables, the yields on bonds are the sum of expected short spot rates and a risk premium for random bond prices. However, in our study, we assume a risk-neutral environment, i.e. zero risk premiums. As tools to deal with stochastic interest rates, various interest rate term structure models are considered. The Vasicek model, the Heath-Jarrow-Morton (hereafter “HJM”) approach and Cairns’ model are explained in detail. The history and nature of the very low interest rate environment in Japan is described in line with the monetary policy framework of the central bank. An unusual interest rate movement in the very low interest rate environment is identified. A modified HJM approach and Cairns’ model are chosen in our study. Cairns’ model is used to graduate the initial yield curve. The HJM approach with a specific volatility function and modified to deal with very low interest rates is used for simulating subsequent developments of the initial yield curve. After the introduction of various concepts needed to investigate a life insurance policy portfolio, we prepare for simulation by collecting information and by fitting parameters to market observations. The Yen swap curve is chosen as a base yield curve. The simulation results show how much initial capital and/or margins are needed in order to avoid the ruin of a portfolio.

Title: Large deviations and their applications in insurance mathematics Author: Lucia Fuchsová Department: Department of Probability and Mathematical Statistics Supervisor: RNDr. Zbyněk Pawlas, Ph.D. Supervisor's e-mail address: Zbynek.Pawlas@mff.cuni.cz Abstract: In the present work we study large deviations theory. We discuss heavy-tailed distributions, which describe the probability of large claim oc- curence. We are interested in the use of large deviations theory in insurance. We simulate claim sizes and their arrival times for Cramér-Lundberg model and first we analyze the probability that ruin happens in dependence on the parameters of our model for Pareto distributed claim size, next we compare ruin probability for other claim size distributions. For real life data we model the probability of large claim size occurence by generalized Pareto distribu- tion. 1

In this thesis, the discrete time risk model with inhomogeneous claims is considered. This model is used for describing the insurer‘s capital and its components: initial capital, premiums received, and claims paid. The main risk measures, ruin probabilities and Gerber-Shiu function, are investigated and recursive formulas are obtained. These formulas give fast and accurate evaluation of the finite time ruin probabilities and Gerber-Shiu function. However, the infinite time investigations require that the Gerber-Shiu function's values for the initial capital equal to 0 must be known. This is slightly more difficult due to the claim inhomogeneity and for this reason a theorem with explicit expression of the infinite time Gerber-Shiu function for a zero initial capital is proposed. However, for the calculation of the infinite time values, some assumption about underlying claim structure must be made. As a solution the cyclically distributed claims are proposed, the algorithms for application of the theorems are given and numerical examples with graphical output are presented. Finally, a special case of discrete time risk model with inhomogeneous claims distributed according geometric law is investigated. In addition to the main results, another discrete time risk model with inhomogeneous claims acquiring rational values is investigated. Two theorems for evaluation of the finite time ruin probabilities are proved and some examples are presented. / Disertaciniame darbe nagrinėjamas diskretaus laiko rizikos modelis su skirtingai pasiskirsčiusiomis žalomis. Šis modelis aprašo draudimo įmonės turtą įtakojančius veiksnius: pradinį kapitalą, gaunamas įmokas, išmokamas žalas. Išvedamos rekursinės formulės, kurių pagalba galima tiksliai ir greitai rasti baigtinio laiko bankroto tikimybių ir Gerber-Shiu funkcijos vertes. Rekursinės formulės taip pat pateikiamos ir begalinio laiko rizikos matams, tačiau nevienodai pasiskirsčiusių žalų atveju iškyla papildomų sunkumų randant bankroto tikimybę ir Gerber-Shiu funkciją, kai pradinis kapitalas lygus 0. Tam įrodoma atskira teorema, tačiau nedarant jokių prielaidų apie žalų pasiskirstymus, apskaičiuoti vertes lengva tikrai nėra. Kaip išeitis pasiūloma cikliškai pasiskirsčiusių žalų struktūra ir pateikiami algoritmai, leidžiantys teoremas pritaikyti praktiškai. Demonstruojant teoremų ir rekursinių formulių veikimą, pateikiami skaitiniai pavyzdžiai su grafinėmis iliustracijomis bei programų kodai. Galiausiai nagrinėjamas atskiras diskretaus laiko rizikos modelio atvejis, kai žalos pasiskirsčiusios skirtingai pagal geometrinį dėsnį. Disertacijoje taip pat yra nagrinėjamas diskretaus laiko rizikos modelis su skirtingai pasiskirsčiusiomis žalomis, kurios įgyja racionalias reikšmes, bei kintančiomis įmokomis ir pradiniu kapitalu, taip pat įgyjančiais racionalias reikšmes su tam tikra sąlyga. Įrodomos dvi teoremos kaip rasti tokio modelio baigtinio laiko bankroto tikimybę ir keli... [toliau žr. visą tekstą]

Disertaciniame darbe nagrinėjamas diskretaus laiko rizikos modelis su skirtingai pasiskirsčiusiomis žalomis. Šis modelis aprašo draudimo įmonės turtą įtakojančius veiksnius: pradinį kapitalą, gaunamas įmokas, išmokamas žalas. Išvedamos rekursinės formulės, kurių pagalba galima tiksliai ir greitai rasti baigtinio laiko bankroto tikimybių ir Gerber-Shiu funkcijos vertes. Rekursinės formulės taip pat pateikiamos ir begalinio laiko rizikos matams, tačiau nevienodai pasiskirsčiusių žalų atveju iškyla papildomų sunkumų randant bankroto tikimybę ir Gerber-Shiu funkciją, kai pradinis kapitalas lygus 0. Tam įrodoma atskira teorema, tačiau nedarant jokių prielaidų apie žalų pasiskirstymus, apskaičiuoti vertes lengva tikrai nėra. Kaip išeitis pasiūloma cikliškai pasiskirsčiusių žalų struktūra ir pateikiami algoritmai, leidžiantys teoremas pritaikyti praktiškai. Demonstruojant teoremų ir rekursinių formulių veikimą, pateikiami skaitiniai pavyzdžiai su grafinėmis iliustracijomis bei programų kodai. Galiausiai nagrinėjamas atskiras diskretaus laiko rizikos modelio atvejis, kai žalos pasiskirsčiusios skirtingai pagal geometrinį dėsnį. Disertacijoje taip pat yra nagrinėjamas diskretaus laiko rizikos modelis su skirtingai pasiskirsčiusiomis žalomis, kurios įgyja racionalias reikšmes, bei kintančiomis įmokomis ir pradiniu kapitalu, taip pat įgyjančiais racionalias reikšmes su tam tikra sąlyga. Įrodomos dvi teoremos kaip rasti tokio modelio baigtinio laiko bankroto tikimybę ir keli... [toliau žr. visą tekstą] / In this thesis, the discrete time risk model with inhomogeneous claims is considered. This model is used for describing the insurer‘s capital and its components: initial capital, premiums received, and claims paid. The main risk measures, ruin probabilities and Gerber-Shiu function, are investigated and recursive formulas are obtained. These formulas give fast and accurate evaluation of the finite time ruin probabilities and Gerber-Shiu function. However, the infinite time investigations require that the Gerber-Shiu function's values for the initial capital equal to 0 must be known. This is slightly more difficult due to the claim inhomogeneity and for this reason a theorem with explicit expression of the infinite time Gerber-Shiu function for a zero initial capital is proposed. However, for the calculation of the infinite time values, some assumption about underlying claim structure must be made. As a solution the cyclically distributed claims are proposed, the algorithms for application of the theorems are given and numerical examples with graphical output are presented. Finally, a special case of discrete time risk model with inhomogeneous claims distributed according geometric law is investigated. In addition to the main results, another discrete time risk model with inhomogeneous claims acquiring rational values is investigated. Two theorems for evaluation of the finite time ruin probabilities are proved and some examples are presented.