Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-11T20:52:36Z No. of bitstreams: 1 ANTONIO LUIZ SOARES SANTOS - DISSERTAÇÃO PPGMAT 2007..pdf: 419776 bytes, checksum: 648bbdd93c2d2fdd67f2dd99256a1ff7 (MD5) / Made available in DSpace on 2018-07-11T20:52:36Z (GMT). No. of bitstreams: 1 ANTONIO LUIZ SOARES SANTOS - DISSERTAÇÃO PPGMAT 2007..pdf: 419776 bytes, checksum: 648bbdd93c2d2fdd67f2dd99256a1ff7 (MD5) Previous issue date: 2007-02 / Neste trabalho estudamos o processo de risco a tempo discreto, considerado modelo clássico na teoria do risco, com variantes propostas por Jun Cai e David Dickson (2004). Serão incluídas taxas de juros, as quais seguem uma Cadeia de Markov, e seus efeitos, em relação à probabilidade de ruína serão analisados. O conhecido limitante superior proposto por Lundberg para essa probabilidade fica reduzido em virtude dessa nova abordagem e a desigualdade clássica é generalizada. / In this work we study discrete time risk process considered classical model, with variants proposed by Jun Cai and David Dickson (2004). Rates of interest, which follows a Markov chain will be introduced and their effect on the ruin probabilities will be analysed. Generalized Lundberg inequalities will be obtained and shown how the classical bounds for the ruin probability can be derived.

Made available in DSpace on 2015-03-03T15:28:30Z (GMT). No. of bitstreams: 1 CarlosAGS_DISSERT.pdf: 446797 bytes, checksum: 18c0fd9fe8336f2abf045fa977920e6c (MD5) Previous issue date: 2010-03-19 / In this work, we present a risk theory application in the following scenario: In each period of time we have a change in the capital of the ensurance company and the outcome of a two-state Markov chain stabilishs if the company pays a benece it heat to one of its policyholders or it receives a Hightimes c > 0 paid by someone buying a new policy. At the end we will determine once again by the recursive equation for expectation the time ruin for this company / Neste trabalho, apresentamos uma aplica??o da teoria do risco com o seguinte cen?rio: as mudan?as no capital de uma seguradora acontecem em cada instante de tempo e o pagamento de uma indeniza??o ou recebimento de um pr?mio ? decidido pelo resultado de uma cadeia de Markov de dois estados. Nesta situa??o calculamos a probabilidade de ru?na e o tempo esperado de ru?na quando o valor da indeniza??o ? um m?tiplo do valor do pr?mio

Made available in DSpace on 2015-03-03T15:28:33Z (GMT). No. of bitstreams: 1 RafaelaHSR_DISSERT.pdf: 2083489 bytes, checksum: 205ac27477b3c2a065fe9cb369c9200e (MD5) Previous issue date: 2013-02-28 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In the work reported here we present theoretical and numerical results about a Risk Model with Interest Rate and Proportional Reinsurance based on the article Inequalities for the ruin probability in a controlled discrete-time risk process by Ros ario Romera and Maikol Diasparra (see [5]). Recursive and integral equations as well as upper bounds for the Ruin Probability are given considering three di erent approaches, namely, classical Lundberg inequality, Inductive approach and Martingale approach. Density estimation techniques (non-parametrics) are used to derive upper bounds for the Ruin Probability and the algorithms used in the simulation are presented / Neste trabalho apresentamos resultados te?ricos e num?ricos referentes a um Modelo de Risco com Taxa de Juros e Resseguro Proporcional baseados no artigo Inequalities for the ruin probability in a controlled discrete-time risk process de Ros?rio Romera e Maikol Diasparra (veja [5]). Equa??es recursivas e integrais para a Probabilidade de Ru?na s?o obtidas bem como cotas superiores para a mesma por diferentes abordagens, a saber, pela cl?ssica desigualdade de Lundberg, pela abordagem Indutiva e pela abordagem Martingale. T?cnicas de estima??o de densidade (n?o-param?tricas) s?o utilizadas para a obten??o das cotas para a Probabilidade de Ru?na e os algoritmos utilizados na simula??o s?o apresentados

保險人因提供最低保證給付之投資型商品，使公司亦涉入投資風險。本研究旨在探討最低保證給付人壽保險附約之風險分析。首先利用隨機模型建構投資者帳戶價值的動態過程，進而推導出在未來時點帳戶發生餘額不足之機率及其所符合的偏微分方程式。並藉由數值方法－有限差分法，求出投資帳戶餘額不足之機率。最終，以不同的參數選取之下，進行敏感度分析，探討參數值的設定對於帳戶發生餘額不足之機率的影響。本研究結果可以提供保險公司與監理機關，作為日後發行保證給付商品時，一項風險管理上的考慮因素。 研究結果可以歸納為兩點結論： 1. 在市場因素中，投資帳戶連結之標的報酬率與帳戶餘額不足機率呈現反向變動，而波動度則是與帳戶餘額不足機率呈現正向變動。在兩因素同時考慮下，當報酬率愈高且波動度愈低，投資帳戶發生餘額不足的機率會愈低。當波動度愈高且報酬率愈低時，帳戶餘額不足機率則會愈高。其兩者的力量會相互抵銷，對投資帳戶餘額不足之機率的影響需視何者的力量較強而定。 2. 在條款設計的因素中，保證附約相關費用率、保證提領比率與保證提領期間對於投資帳戶發生餘額不足機率的影響皆呈現正向的關係。而投資帳戶期初的價值則與帳戶餘額不足機率呈現反向變動。其中保證提領比率對於投資帳戶的價值影響最大，其帳戶餘額不足機率之變動百分比相較於其他因素而言，變動幅度較大，範圍皆大於4%以上，甚至高達37.11%。 / Insurers have investment risks because they issue the guaranteed minimum benefit life insurance riders. Therefore, the purpose of this thesis is analyzing the risk for the riders. In the context, we implement numerical PDE solution to compute the ruin probability of separate account which is the probability that guaranteed minimum benefit life insurance riders will lead to financial insolvency under stochastic investment returns. Moreover, we will do sensitivity analyses to discuss the two aspects, market factors and contract designs, how to influence the ruin probability. Finally, we conclude two main results: 1. For market factors, the rate of investment return is negatively related to ruin probability; however, the volatility is positive correlation. 2. For contract designs, the results show negative correlation between ruin probability and insurance fee, withdrawals, and withdrawal period. But the initial account value shows positive correlation.

Two optimisation problems were considered, in which market exposure is indirectly controlled. The first one models the capital of a company and an independent portfolio of new businesses, each one represented by a Cram\'r-Lundberg process. The company can choose the proportion of new business it wants to take on and can alter this proportion over time. Here the objective is to find a strategy that maximises the survival probability. We use a point processes framework to deal with the impact of an adapted strategy in the intensity of the new business. We prove that when Cram\'{e}r-Lundberg processes with exponentially distributed claims, it is optimal to choose a threshold type strategy, where the company switches between owning all new businesses or none depending on the capital level. For this type of processes that change both drift and jump measure when crossing the constant threshold, we solve the one and two-sided exit problems. This optimisation problem is also solved when the capital of the company and the new business are modelled by spectrally positive L\'vy processes of bounded variation. Here the one-sided exit problem is solved and we prove optimality of the same type of threshold strategy for any jump distribution. The second problem is a stochastic variation of the work done by Taylor about underwriting in a competitive market. Taylor maximised discounted future cash flows over a finite time horizon in a discrete time setting when the change of exposure from one period to the next has a multiplicative form involving the company's premium and the market average premium. The control is the company's premium strategy over a the mentioned finite time horizon. Taylor's work opened a rich line of research, and we discuss some of it. In contrast with Taylor's model, we consider the market average premium to be a Markov chain instead of a deterministic vector. This allows to model uncertainty in future conditions of the market. We also consider an infinite time horizon instead of finite. This solves the time dependency in Taylor's optimal strategies that were giving unrealistic results. Our main result is a formula to calculate explicitly the value function of a specific class of pricing strategies. Further we explore concrete examples numerically. We find a mix of optimal strategies where in some examples the company should follow the market while in other cases should go against it.

動態規劃的問題並不一定都存在封閉解(closed form solution)，即使存在，其過程往往也相當繁雜。本研究擬以 Gerrard & Haberman (2004) 的模型為基礎，並使用逼近動態規劃理論解的數值方法來求解，此方法參考自黃迪揚(2009)，其研究探討在有無封閉解的動態規劃下，使用此數值方法求解可以得到 逼近解。本篇嘗試延伸其方法，針對不同類型的限制，做更多不同的變化。Gerrard & Haberman (2004)推導出退休後投資於風險性資產與無風險性資產之最適投資策略封閉解， 本研究欲將模型投資之兩資產衍生至三資產，分別投資在高風險資產、中風險資產與無風險資產，實際市場狀況下禁止買空賣空的情況與風險趨避程度限制資產投資比例所造成的影響。並探討兩資產與三資產下的投資結果，並加入不同的目標函數:使用控制變異數的限制式來降低破產機率、控制帳戶差異部位讓投資更具效率性。雖然加入這些限制式會導致目標函 數過於複雜，但是用此數值方法還是可以得出逼近解。 / Dynamic Programming’s solution is not always a closed form. If it do exist, the solution of progress may be too complicated. Our research is based on the investing model in Gerrard & Haberman (2004), using the numerical solution by Huang (2009) to solve the dynamic programming problem. In his research, he found out that whether dynamic programming problem has the closed form, using the numerical solution to solve the problems, which could get similar result. So in our research, we try to use this solution to solve more complicate problems. Gerrard & Haberman (2004) derived the closed form solution of optimal investing strategy in post retirement investment plan, investing in risky asset and riskless asset. In this research we try to invest in three assets, investing in high risk asset, middle risk asset and riskless asset. Forbidden short buying and short selling, how risk attitude affect investment behavior in risky asset and riskless asset. We also observe the numerical result of 2 asset and 3 asset, using different objective functions : using variance control to avoid ruin risk, consideration the distance between objective account and actual account to improve investment effective. Although using these restricts may increase the complication of objective functions, but we can use this numerical solution to get the approximating solution.

In this work we describe the basic facts of non-life insurance and then explain risk processes. In particular, we will explain in detail the asymptotic behavior of the probability that an insurance product may end up in ruin during its lifetime. As expected, the behavior of such asymptotic probability will be highly dependent on the tail distribution of each claim. / En este artículo describimos los conceptos básicos relacionados a seguros que no sean de vida y luego explicamos procesos de riesgo. En particular, tratamos al detalle el comportamiento asintótico de la probabilidad de que un producto sea declarado en ruina. Como es suponible, el comportamiento en el horizonte depende de la cola de la distribución de las primas.

In this thesis, nonlinearly perturbed stochastic models in discrete time are considered. We give algorithms for construction of asymptotic expansions with respect to the perturbation parameter for various quantities of interest. In particular, asymptotic expansions are given for solutions of renewal equations, quasi-stationary distributions for semi-Markov processes, and ruin probabilities for risk processes. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 4: Manuscript. Paper 5: Manuscript. Paper 6: Manuscript.</p>

Nous y introduisons une nouvelle classe de distributions bivariées de type Marshall-Olkin, la distribution Erlang bivariée. La transformée de Laplace, les moments et les densités conditionnelles y sont obtenus. Les applications potentielles en assurance-vie et en finance sont prises en considération. Les estimateurs du maximum de vraisemblance des paramètres sont calculés par l'algorithme Espérance-Maximisation. Ensuite, notre projet de recherche est consacré à l'étude des processus de risque multivariés, qui peuvent être utiles dans l'étude des problèmes de la ruine des compagnies d'assurance avec des classes dépendantes. Nous appliquons les résultats de la théorie des processus de Markov déterministes par morceaux afin d'obtenir les martingales exponentielles, nécessaires pour établir des bornes supérieures calculables pour la probabilité de ruine, dont les expressions sont intraitables. / In this contribution, we introduce a new class of bivariate distributions of Marshall-Olkin type, called bivariate Erlang distributions. The Laplace transform, product moments and conditional densities are derived. Potential applications of bivariate Erlang distributions in life insurance and finance are considered. Further, our research project is devoted to the study of multivariate risk processes, which may be useful in analyzing ruin problems for insurance companies with a portfolio of dependent classes of business. We apply results from the theory of piecewise deterministic Markov processes in order to derive exponential martingales needed to establish computable upper bounds of the ruin probabilities, as their exact expressions are intractable.

Nous y introduisons une nouvelle classe de distributions bivariées de type Marshall-Olkin, la distribution Erlang bivariée. La transformée de Laplace, les moments et les densités conditionnelles y sont obtenus. Les applications potentielles en assurance-vie et en finance sont prises en considération. Les estimateurs du maximum de vraisemblance des paramètres sont calculés par l'algorithme Espérance-Maximisation. Ensuite, notre projet de recherche est consacré à l'étude des processus de risque multivariés, qui peuvent être utiles dans l'étude des problèmes de la ruine des compagnies d'assurance avec des classes dépendantes. Nous appliquons les résultats de la théorie des processus de Markov déterministes par morceaux afin d'obtenir les martingales exponentielles, nécessaires pour établir des bornes supérieures calculables pour la probabilité de ruine, dont les expressions sont intraitables. / In this contribution, we introduce a new class of bivariate distributions of Marshall-Olkin type, called bivariate Erlang distributions. The Laplace transform, product moments and conditional densities are derived. Potential applications of bivariate Erlang distributions in life insurance and finance are considered. Further, our research project is devoted to the study of multivariate risk processes, which may be useful in analyzing ruin problems for insurance companies with a portfolio of dependent classes of business. We apply results from the theory of piecewise deterministic Markov processes in order to derive exponential martingales needed to establish computable upper bounds of the ruin probabilities, as their exact expressions are intractable.