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

A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium

In a general Sparre Andersen risk model with surplus-dependent premium income, the generalization of Gerber-Shiu function proposed by Cheung et al. (2010a) is studied. A general expression for such Gerber-Shiu function is derived, and it is shown that its determination reduces to the evaluation of a transition function which is independent of the penalty function. Properties of and explicit expressions for such a transition function are derived when the surplus process is subject to (i) constant premium; (ii) a threshold dividend strategy; or (iii) credit interest. Extension of the approach is discussed for an absolute ruin model with debit interest. © 2011 Elsevier B.V.postprin

On the discounted aggregate claim costs until ruin in dependent Sparre Andersen risk processes

In this paper, a dependent Sparre Andersen risk process in which the joint density of the interclaim time and the resulting claim severity satisfies the factorization as in Willmot and Woo (2012) is considered. We study a generalization of the Gerber-Shiu function (i) whose penalty function further depends on the surplus level immediately after the second last claim before ruin (Cheung et al. (2010a)); and (ii) which involves the moments of the discounted aggregate claim costs until ruin. The generalized discounted density with a moment-based component proposed in Cheung (2013) plays a key role in deriving recursive defective renewal equations. We pay special attention to the case where the marginal distribution of the interclaim times is Coxian, and the required components in the recursion are obtained. A reverse type of dependency structure where the claim severities follow a combination of exponentials is also briefly discussed, and this leads to a nice explicit expression for the expected discounted aggregate claims until ruin. Our results are applied to generate some numerical examples involving (i) the covariance of the time of ruin and the discounted aggregate claims until ruin; and linebreak (ii) the expectation, variance and third central moment of the discounted aggregate claims until ruin.postprin

On the sparre - Andersen risk model with different type of interclaim times distributions

Doutoramento em Matemática Aplicada à Economia e GestãoThe inevitability of taxes and regulations, that cause agents to go underground, forces the authorities to tolerate some underground economic activity and grants the underground economy natural features. The natural level of the underground economy is defined as the level of underground economic activity in the decentralized equilibrium, provided that the actual structural characteristics of the economy and social preferences are accounted for by imbedding them in the Walrasian system of general equilibrium equations. Its existence is proven using two variants of neoclassical general equilibrium models. The underground economy is found to influence the successfulness of fiscal consolidation programmes, depending on the position of the economy relative to critical fiscal thresholds associated with the natural level of the underground economy. Tax increases yield higher tax proceeds up to the threshold, and lower tax proceeds, passed the threshold, due to a stronger expansion of the natural level of the underground economy. Tax proceeds reach their maximum at the threshold. Tax based programmes are found ineffective in high tax developed economies, operating passed the threshold. In contrast, its successfulness in the developing world, where most economies operate below the threshold with low taxes, is not influenced by the underground economy.Perante a inevitabilidade de impostos e regulamentação, que estão na origem da economia subterrânea, as autoridades vêem-se forçadas a tolerar actividades económicas subterrâneas. Isto confere um carácter natural à economia subterrânea. A existência de uma taxa natural de economia subterrânea é provado utilizando dois modelos neoclássicos de equilíbrio gereal. A taxa natural de economia subterrânea define-se como o nível de actividade económica subterrânea no equilíbrio descentralizado, dadas as propriedades estruturais da economia e das preferências sociais, que se incluem no sistema Walrasiano de equações de equilíbrio geral. Prova-se que a economia subterrânea influencia o resultado de programas de consolidação orçamental. Isto depende da localização da economia face a valores fiscais críticos associados à taxa natural de economia subterrânea. A seguir a um aumento de impostos, as receitas começam por crescer, atingindo o máximo no ponto crítico, para a seguir cair, devido a uma expansão da taxa natural de economia subterrânea. Programas assentes no aumento de impostos não são bem sucedidos em países desenvolvidos com cargas fiscais elevadas, que operam além do ponto crítico. Já os países em desenvolvimento, cuja maioria opera abaixo do ponto crítico com cargas fiscais baixas, a economia subterrânea não parece influenciar a eficácia dos programas. Palavras chave: Economia subterrânea, evasão de impostos, politica orçamental, consolidação orçamental, defice, divida, modelos de equilíbrio geral aplicados, modelos de crescimento economico de dois sectores, simulação.N/

The Effect of a Threshold Proportional Reinsurance Strategy on Ruin Probabilities

In the context of a compound Poisson risk model, we define a threshold proportional reinsurance strategy: A retention level k1 is applied whenever the reserves are less than a determinate threshold b, and a retention level k2 is applied in the other case. We obtain the integro-differential equation for the Gerber-Shiu function (defined in Gerber and Shiu (1998)) in this model, which allows us to obtain the expressions for ruin probability and Laplace transforms of time of ruin for several distributions of the claim sizes. Finally, we present some numerical results.time of ruin, threshold proportional reinsurance strategy, ruin probability, gerber-shiu function

A note on deficit analysis in dependency models involving Coxian claim amounts

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An introduction to Gerber-Shiu analysis

A valuable analytical tool to understand the event of ruin is a Gerber-Shiu discounted penalty function. It acts as a unified means of identifying ruin-related quantities which may help insurers understand their vulnerability ruin. This thesis provides an introduction to the basic concepts and common techniques used for the Gerber-Shiu analysis. Chapter 1 introduces the insurer's surplus process in the ordinary Sparre Andersen model. Defective renewal equations, the Dickson-Hipp transform, and Lundberg's fundamental equation are reviewed. Chapter 2 introduces the classical Gerber-Shiu discounted penalty function. Two framework equations are derived by conditioning on the first drop in surplus below its initial value, and by conditioning on the time and amount of the first claim. A detailed discussion is provided for each of these conditioning arguments. The classical Poisson model (where interclaim times are exponentially distributed) is then considered. We also consider when claim sizes are exponentially distributed. Chapter 3 introduces the Gerber-Shiu function in the delayed renewal model which allows the time until the first claim to be distributed differently than subsequent interclaim times. We determine a functional relationship between the Gerber-Shiu function in the ordinary Sparre Andersen model and the Gerber-Shiu function in the delayed model for a class of first interclaim time densities which includes the equilibrium density for the stationary renewal model, and the exponential density. To conclude, Chapter 4 introduces a generalized Gerber-Shiu function where the penalty function includes two additional random variables: the minimum surplus level before ruin, and the surplus immediately after the claim before the claim causing ruin. This generalized Gerber-Shiu function allows for the study of random variables which otherwise could not be studied using the classical definition of the function. Additionally, it is assumed that the size of a claim is dependant on the interclaim time that precedes it. As is done in Chapter 2, a detailed discussion of each of the two conditioning arguments is provided. Using the uniqueness property of Laplace transforms, the form of joint defective discounted densities of interest are determined. The classical Poisson model and the exponential claim size assumption is also revisited

Gerber-Shiu analysis in some dependent Sparre Andersen risk models

In this thesis, we consider a generalization of the classical Gerber-Shiu function in various risk models. The generalization involves introduction of two new variables in the original penalty function including the surplus prior to ruin and the deficit at ruin. These new variables are the minimum surplus level before ruin occurs and the surplus immediately after the second last claim before ruin occurs. Although these quantities can not be observed until ruin occurs, we can still identify their distributions in advance because they do not functionally depend on the time of ruin, but only depend on known quantities including the initial surplus allocated to the business. Therefore, some ruin related quantities obtained by incorporating four variables in the generalized Gerber-Shiu function can help our understanding of the analysis of the random walk and the resultant risk management. In Chapter 2, we demonstrate the generalized Gerber-Shiu functions satisfy the defective renewal equation in terms of the compound geometric distribution in the ordinary Sparre Andersen renewal risk models (continuous time). As a result, forms of joint and marginal distributions associated with the variables in the generalized penalty function are derived for an arbitrary distribution of interclaim/interarrival times. Because the identification of the compound geometric components is difficult without any specific conditions on the interclaim times, in Chapter 3 we consider the special case when the interclaim time distribution is from the Coxian class of distribution, as well as the classical compound Poisson models. Note that the analysis of the generalized Gerber-Shiu function involving three (the classical two variables and the surplus after the second last claim) is sufficient to study of four variable. It is shown to be true even in the cases where the interclaim of the first event is assumed to be different from the subsequent interclaims (i.e. delayed renewal risk models) in Chapter 4 or the counting (the number of claims) process is defined in the discrete time (i.e. discrete renewal risk models) in Chapter 5. In Chapter 6 the two-sided bounds for a renewal equation are studied. These results may be used in many cases related to the various ruin quantities from the generalized Gerber-Shiu function analyzed in previous chapters. Note that the larger number of iterations of computing the bound produces the closer result to the exact value. However, for the nonexponential bound the form of bound contains the convolution involving usually heavy-tailed distribution (e.g. heavy-tailed claims, extreme events), we need to find the alternative method to reinforce the convolution computation in this case