Risk models with dependence and perturbation

In ruin theory, the surplus process of an insurance company is usually modeled by the classical compound Poisson risk model or its general version, the Sparre-Andersen risk model. Under these models, the claim amounts and the inter-claim times are assumed to be independently distributed, which is not always appropriate in practice. In recent years, risk models relaxing the independence assumption have drawn increasing attention. However, previous research mostly considers the so call dependent Sparre-Andersen risk model under which the pairs of random variables consisting of the inter-claim time and the next claim amount remain independent of each other. In this thesis, we aim to examine the opposite case. Namely, the distribution of the time until the next claim depends on the size of the previous claim amount. Explicit solutions for the Gerber-Shiu function are provided for arbitrary claim sizes and various ruin-related quantities are obtained as special cases. Numerical examples are also presented. The dependent insurance risk process is further generalized to a perturbed version to incorporate small fluctuations of the underlying surplus process. Explicit solutions for the Gerber-Shiu funtion are deduced along with applications and examples. Lastly, we introduce a perturbed dependence structure into the dual risk model and study the ruin time problem. Exact solutions for the Laplace transform and the first moment of the time to ruin with an arbitrary gain-size distribution are obtained. Applications with numerical examples are provided to illustrate the impact of the dependence structure and the perturbation

On the Dual Risk Models

Abstract This thesis focuses on developing and computing ruin-related quantities that are potentially measurements for the dual risk models which was proposed to describe the annuity-type businesses from the perspective of the collective risk theory in 1950’s. In recent years, the dual risk models are revisited by many researchers to quantify the risk of the similar businesses as the annuity-type businesses. The major extensions included in this thesis consist of two aspects: the first is to search for new ruin-related quantities that are potentially indices of the risk for well-established dual models; the other aspect is to generalize the settings of the dual models instead of the ruin quantities. There are four separate articles in this thesis, in which the first (Chapter 2) and the last (Chapter 5) belong to the first type of extensions while the others (Chapter 3 and Chapter 4) belong to the generalizations of the dual models. The first article (Chapter 2) studies the discounted moments of the surplus at the time of the last jump before ruin for the compound Poisson dual risk model. The idea comes from that the ruin of the compound Poisson dual models is caused by absence of positive jumps within a period with length being propotional to the surplus at the time of the last jump. As a quantity related to a non-stopping time, the explicit expression of the target quantity is obtained through integro-differential equations. The second article (Chapter 3) investigate the Sparre-Andersen dual risk models in which the epochs are independently, identically distributed generalized Erlang-n random variables. An important difference between this model and some other models such as the Erlang-n dual risk models is that the roots to the generalized Lundberg’s equation are not necessarily distinct. By taking the multiple roots into account, the explicit expressions of the Laplace transform of the time to ruin and expected discounted aggregate dividends under the threshold strategy and exponential distributed revenues are derived. The third article (Chapter 4) revisits the the dual Lévy risk model. The target ruin quantity is the expected discounted aggregate dividends paid up to ruin under the threshold dividend strategy. The explicit expression is obtained in terms of the q-scale functions through constructing a new dividend strategy having the target ruin quantity converging to that under the threshold strategy. Also, the optimality of the threshold strategy among all the absolutely continuous stategies when evaluating the target quantity as a value function is discussed. The fourth article (Chapter 5) initiate the study of the Parisian ruin problem for the general dual Lévy risk models. Unlike the regular ruin for the dual models, the deficit at Parisian ruin is not necessarily equal to zero. Hence we introduce the Gerber-Shiu expected discounted penalty function (EPDF) at the Parisian ruin and obtain an explicit expression for this function. Keywords: Sparre-Andersen dual models, expected discounted aggregate dividends, dual Levy risk models, Parisian ruin, Gerber-Shiu function ii

On the Sparre-Andersen Risk Models

This thesis develops several strategies for calculating ruin-related quantities for a variety of extended risk models. We focus on the Sparre-Andersen risk model, also known as the renewal risk model. The idea of arbitrary distribution for the waiting time between claim payments arose in the 1950’s from the collective risk theory, and received many extensions and modifications in recent years. Our goal is to tackle model assumptions that are either too relaxed for traditional methods to apply, or so complicated that elaborate algebraic tools are needed to obtain explicit solutions. In Chapter 2, we consider a Lévy risk process and a Sparre-Andersen risk process with Parisian ruin in the presence of a constant dividend barrier. We demonstrate that with few exceptions, ruin occurs with certainty. Generalizations to certain dependent risk processes are discussed. We also provide a reinsurance contract in which the certainty of ruin can be avoided. In Chapter 3, we investigate a class of Sparre-Andersen risk processes in which the inter-claim time is rational-distributed. A key property of the rational class is derived, which allows for direct derivation of an integro-differential equation satisfied by a probability concerning the maximum surplus. The solution is constructed using a set of linearly independent functions, one of which is obtained by a standard technique through a defective renewal equation while the rest are obtained via a homogeneous equation. The necessary boundary conditions are presented. We also provide examples involving rational claim sizes as well as an application to the total dividends paid under a threshold strategy. In Chapter 4, we extend an exponential-combination dependence structure to an Erlang-combination for the Sparre-Andersen risk models in presence of diffusion. A set of tools are developed for establishing certain integro-differential equations in Gerber–Shiu analysis. This new technique lifts previous constraint on the multiplicities of parameters of the inter-claim times. We then illustrate applications of these equations under a variety of special dependence models. Results are compared with existing literature, including the diffusion-free cases. Finally, in Chapter 5, we collect various results and provide conclusions. We also give an outline of potential future research

Analysis of some risk models involving dependence

The seminal paper by Gerber and Shiu (1998) gave a huge boost to the study of risk theory by not only unifying but also generalizing the treatment and the analysis of various risk-related quantities in one single mathematical function - the Gerber-Shiu expected discounted penalty function, or Gerber-Shiu function in short. The Gerber-Shiu function is known to possess many nice properties, at least in the case of the classical compound Poisson risk model. For example, upon the introduction of a dividend barrier strategy, it was shown by Lin et al. (2003) and Gerber et al. (2006) that the Gerber-Shiu function with a barrier can be expressed in terms of the Gerber-Shiu function without a barrier and the expected value of discounted dividend payments. This result is the so-called dividends-penalty identity, and it holds true when the surplus process belongs to a class of Markov processes which are skip-free upwards. However, one stringent assumption of the model considered by the above authors is that all the interclaim times and the claim sizes are independent, which is in general not true in reality. In this thesis, we propose to analyze the Gerber-Shiu functions under various dependent structures. The main focus of the thesis is the risk model where claims follow a Markovian arrival process (MAP) (see, e.g., Latouche and Ramaswami (1999) and Neuts (1979, 1989)) in which the interclaim times and the claim sizes form a chain of dependent variables. The first part of the thesis puts emphasis on certain dividend strategies. In Chapter 2, it is shown that a matrix form of the dividends-penalty identity holds true in a MAP risk model perturbed by diffusion with the use of integro-differential equations and their solutions. Chapter 3 considers the dual MAP risk model which is a reflection of the ordinary MAP model. A threshold dividend strategy is applied to the model and various risk-related quantities are studied. Our methodology is based on an existing connection between the MAP risk model and a fluid queue (see, e.g., Asmussen et al. (2002), Badescu et al. (2005), Ramaswami (2006) and references therein). The use of fluid flow techniques to analyze risk processes opens the door for further research as to what types of risk model with dependency structure can be studied via probabilistic arguments. In Chapter 4, we propose to analyze the Gerber-Shiu function and some discounted joint densities in a risk model where each pair of the interclaim time and the resulting claim size is assumed to follow a bivariate phase-type distribution, with the pairs assumed to be independent and identically distributed (i.i.d.). To this end, a novel fluid flow process is constructed to ease the analysis. In the classical Gerber-Shiu function introduced by Gerber and Shiu (1998), the random variables incorporated into the analysis include the time of ruin, the surplus prior to ruin and the deficit at ruin. The later part of this thesis focuses on generalizing the classical Gerber-Shiu function by incorporating more random variables into the so-called penalty function. These include the surplus level immediately after the second last claim before ruin, the minimum surplus level before ruin and the maximum surplus level before ruin. In Chapter 5, the focus will be on the study of the generalized Gerber-Shiu function involving the first two new random variables in the context of a semi-Markovian risk model (see, e.g., Albrecher and Boxma (2005) and Janssen and Reinhard (1985)). It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation, and some discounted joint densities involving the new variables are derived. Chapter 6 revisits the MAP risk model in which the generalized Gerber-Shiu function involving the maximum surplus before ruin is examined. In this case, the Gerber-Shiu function no longer satisfies a defective renewal equation. Instead, the generalized Gerber-Shiu function can be expressed in terms of the classical Gerber-Shiu function and the Laplace transform of a first passage time that are both readily obtainable. In a MAP risk model, the interclaim time distribution must be phase-type distributed. This leads us to propose a generalization of the MAP risk model by allowing for the interclaim time to have an arbitrary distribution. This is the subject matter of Chapter 7. Chapter 8 is concerned with the generalized Sparre Andersen risk model with surplus-dependent premium rate, and some ordering properties of certain ruin-related quantities are studied. Chapter 9 ends the thesis by some concluding remarks and directions for future research

On dividends and other quantities of interest in the dual risk model

Doutoramento em Matemática Aplicada à Economia e à GestãoNesta dissertação trabalhamos em teoria do risco. Damos principal ênfase principal nos modelos de risco e teoria de ruína, dedicando a nossa atenção a algumas das mais interessantes e relevantes quantidades da área: a probabilidade da ruína, a transformada de Laplace e os dividendos descontados esperados. Os modelos de risco têm o objetivo de resolver, ou pelo menos, fornecer uma solução aproximada, a problemas que aparecem na prática do negócio dos seguros. Os desenvolvimentos que produzimos nesta dissertação têm a mesma finalidade. A nossa intenção é apresentar novas ferramentas para o cálculo das quantidades mencionadas acima, e uma melhor compreensão delas na prática. Consideramos o modelo dual de risco quando os tempos entre ganhos seguem uma distribuição exponencial matricial e, quando for possível, dar exemplos dos nossos resultados para casos particulares, como as distribuições Phase–Type e Erlang. Mostramos, na maioria dos casos, fórmulas e fazemos uso de técnicas matemáticas de várias áreas, como a teoria da probabilidade, a teoria das equações integro–diferenciais, a ágebra linear, análise complexa, entre outras.In this manuscript we work on risk theory. The main emphasis is on risk models and ruin theory, devoting our attention to some of the most interesting and relevant quantities in this area: ruin probabilities, Laplace transforms and expected discounted dividends. Risk models are meant to solve or, at least, provide an approximate solution, to problems that appear in the practice of the insurance business. The developments we produce in this dissertation have the same goal. Our aim is to present new tools for computation of the quantities mentioned above, and a better understanding of them in the practice. We consider the dual risk model when the interclaim times follow a matrix exponential distribution and, whenever possible, we give examples of our findings for particular cases, like the Phase–Type, the Generalized Erlang and the Erlang distributions. We show, in most cases, explicit formulas and we make use of mathematical techniques from several areas, like probability theory, the theory of integro–differential equations, linear algebra, complex analysis, among others.info:eu-repo/semantics/publishedVersio

On First Passage Time Related Problems for Some Insurance Risk Processes

For many decades, the study of ruin theory has long been one of the central topics of interest in insurance risk management. Research in this area has largely focused on analyzing the insurer’s solvency risk, which is essentially a standard first passage time problem. To model the manner in which the claim experience develops over time for a block of insurance business, various stochastic processes have been proposed and studied. Following the pioneer works of Lundberg [88] and Cramér [32], in which the classical compound Poisson model was proposed to model the insurer’s surplus process, there has been a considerable amount of effort devoted to constructing more realistic risk models to better characterize some practical features of the insurer’s surplus cash flows. This thesis aims to contribute to this line of research and enhance our general understanding of an insurer’s solvency risk. In most analyses of the main risk processes in risk theory, the income process is modelled by a deterministic process which accrues at a constant rate per time unit. As we know, this is a rather simplifying assumption which is far from being realistic in the insurance world, but one under which the solvency risk is typically assessed. To investigate the impact of income processes exhibiting a higher degree of variability on an insurer’s solvency risk, the first part of the thesis focuses on analyzing risk models with random income processes. In Chapter 2, we consider a generalized Sparre Andersen risk model with a random income process which renews at claim instants. Under the setting of this particular generalization of the Sparre Andersen risk model, we investigate the impact of income processes on both infinite-time and finite-time ruin quantities. In Chapter 3, we further extend the results of the risk model proposed in Chapter 2 by analyzing a renewal insurance risk model with two-sided jumps and a random income process. Another class of risk models that has drawn considerable interest in risk theory are the spectrally negative Lévy processes. Thanks to the development of the fluctuation theory of Lévy processes, first passage time analysis of Lévy insurance risk models has flourished in the last two decades, both in terms of models proposed and quantities analyzed. For example, risk models with dividends (or tax) payouts and exotic ruin have received considerable attention in the field of insurance mathematics. Leveraging the extensive literature on fluctuation identities for spectrally negative Lévy processes, the second part of the thesis considers some first passage problems in this context. In Chapter 4, we study a refracted Lévy risk model with delayed dividend pullbacks and obtain explicit expressions for two-sided exit identities for the proposed risk process. Chapter 5 introduces two types of random times with the goal of bridging the first and the last passage times’ analyses. The Laplace transforms of these two random times are derived for the class of spectrally negative Lévy processes. To ensure that the thesis flows smoothly, Chapter 1 introduces the background literature and main motivations of this thesis and provides the relevant mathematical preliminaries for the later chapters. Chapter 6 concludes the thesis with some remarks and potential directions for future research

The Gerber-Shiu Expected Penalty Function for the Risk Model with Dependence and a Constant Dividend Barrier

We consider a compound Poisson risk model with dependence and a constant dividend barrier. A dependence structure between the claim amount and the interclaim time is introduced through a Farlie-Gumbel-Morgenstern copula. An integrodifferential equation for the Gerber-Shiu discounted penalty function is derived. We also solve the integrodifferential equation and show that the solution is a linear combination of the Gerber-Shiu function with no barrier and the solution of an associated homogeneous integrodifferential equation