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.

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.

Following the introduction of the discounted penalty function by Gerber and Shiu (1998), significant progress has been made on the analysis of various ruin-related quantities in risk theory. As we know, the discounted penalty function not only provides a systematic platform to jointly analyze various quantities of interest, but also offers the convenience to extract key pieces of information from a risk management perspective. For example, by eliminating the penalty function, the Gerber-Shiu function becomes the Laplace-Stieltjes transform of the time to ruin, inversion of which results in a series expansion for the associated density of the time to ruin (see, e.g., Dickson and Willmot (2005)). In this thesis, we propose to analyze the long-standing finite-time ruin problem by incorporating the number of claims until ruin into the Gerber-Shiu analysis. As will be seen in Chapter 2, many nice analytic properties of the original Gerber-Shiu function are preserved by this generalized analytic tool. For instance, the Gerber-Shiu function still satisfies a defective renewal equation and can be generally expressed in terms of some roots of Lundberg's generalized equation in the Sparre Andersen risk model. In this thesis, we propose not only to unify previous methodologies on the study of the density of the time to ruin through the use of Lagrange's expansion theorem, but also to provide insight into the nature of the series expansion by identifying the probabilistic contribution of each term in the expansion through analysis involving the distribution of the number of claims until ruin. In Chapter 3, we study the joint generalized density of the time to ruin and the number of claims until ruin in the classical compound Poisson risk model. We also utilize an alternative approach to obtain the density of the time to ruin based on the Lagrange inversion technique introduced by Dickson and Willmot (2005). In Chapter 4, relying on the Lagrange expansion theorem for analytic inversion, the joint density of the time to ruin, the surplus immediately before ruin and the number of claims until ruin is examined in the Sparre Andersen risk model with exponential claim sizes and arbitrary interclaim times. To our knowledge, existing results on the finite-time ruin problem in the Sparre Andersen risk model typically involve an exponential assumption on either the interclaim times or the claim sizes (see, e.g., Borovkov and Dickson (2008)). Among the few exceptions, we mention Dickson and Li (2010, 2012) who analyzed the density of the time to ruin for Erlang-n interclaim times. In Chapter 5, we propose a significant breakthrough by utilizing the multivariate version of Lagrange's expansion theorem to obtain a series expansion for the density of the time to ruin under a more general distribution assumption, namely when interclaim times are distributed as a combination of n exponentials. It is worth emphasizing that this technique can also be applied to other areas of applied probability. For instance, the proposed methodology can be used to obtain the distribution of some first passage times for particular stochastic processes. As an illustration, the duration of a busy period in a queueing risk model will be examined. Interestingly, the proposed technique can also be used to analyze some first passage times for the compound Poisson processes with diffusion. In Chapter 6, we propose an extension to Kendall's identity (see, e.g., Kendall (1957)) by further examining the distribution of the number of jumps before the first passage time. We show that the main result is particularly relevant to enhance our understanding of some problems of interest, such as the finite-time ruin probability of a dual compound Poisson risk model with diffusion and pricing barrier options issued on an insurer's stock price. Another closely related quantity of interest is the so-called occupation times of the surplus process below zero (also referred to as the duration of negative surplus, see, e.g., Egidio dos Reis (1993)) or in a certain interval (see, e.g., Kolkovska et al. (2005)). Occupation times have been widely used as a contingent characteristic to develop advanced derivatives in financial mathematics. In risk theory, it can be used as an important risk management tool to examine the overall health of an insurer's business. The main subject matter of Chapter 7 is to extend the analysis of occupation times to a class of renewal risk processes. We provide explicit expressions for the duration of negative surplus and the double-barrier occupation time in terms of their Laplace-Stieltjes transform. In the process, we revisit occupation times in the content of the classical compound Poisson risk model and examine some results proposed by Kolkovska et al. (2005). Finally, some concluding remarks and discussion of future research are made in Chapter 8.

Following the introduction of the discounted penalty function by Gerber and Shiu (1998), significant progress has been made on the analysis of various ruin-related quantities in risk theory. As we know, the discounted penalty function not only provides a systematic platform to jointly analyze various quantities of interest, but also offers the convenience to extract key pieces of information from a risk management perspective. For example, by eliminating the penalty function, the Gerber-Shiu function becomes the Laplace-Stieltjes transform of the time to ruin, inversion of which results in a series expansion for the associated density of the time to ruin (see, e.g., Dickson and Willmot (2005)). In this thesis, we propose to analyze the long-standing finite-time ruin problem by incorporating the number of claims until ruin into the Gerber-Shiu analysis. As will be seen in Chapter 2, many nice analytic properties of the original Gerber-Shiu function are preserved by this generalized analytic tool. For instance, the Gerber-Shiu function still satisfies a defective renewal equation and can be generally expressed in terms of some roots of Lundberg's generalized equation in the Sparre Andersen risk model. In this thesis, we propose not only to unify previous methodologies on the study of the density of the time to ruin through the use of Lagrange's expansion theorem, but also to provide insight into the nature of the series expansion by identifying the probabilistic contribution of each term in the expansion through analysis involving the distribution of the number of claims until ruin. In Chapter 3, we study the joint generalized density of the time to ruin and the number of claims until ruin in the classical compound Poisson risk model. We also utilize an alternative approach to obtain the density of the time to ruin based on the Lagrange inversion technique introduced by Dickson and Willmot (2005). In Chapter 4, relying on the Lagrange expansion theorem for analytic inversion, the joint density of the time to ruin, the surplus immediately before ruin and the number of claims until ruin is examined in the Sparre Andersen risk model with exponential claim sizes and arbitrary interclaim times. To our knowledge, existing results on the finite-time ruin problem in the Sparre Andersen risk model typically involve an exponential assumption on either the interclaim times or the claim sizes (see, e.g., Borovkov and Dickson (2008)). Among the few exceptions, we mention Dickson and Li (2010, 2012) who analyzed the density of the time to ruin for Erlang-n interclaim times. In Chapter 5, we propose a significant breakthrough by utilizing the multivariate version of Lagrange's expansion theorem to obtain a series expansion for the density of the time to ruin under a more general distribution assumption, namely when interclaim times are distributed as a combination of n exponentials. It is worth emphasizing that this technique can also be applied to other areas of applied probability. For instance, the proposed methodology can be used to obtain the distribution of some first passage times for particular stochastic processes. As an illustration, the duration of a busy period in a queueing risk model will be examined. Interestingly, the proposed technique can also be used to analyze some first passage times for the compound Poisson processes with diffusion. In Chapter 6, we propose an extension to Kendall's identity (see, e.g., Kendall (1957)) by further examining the distribution of the number of jumps before the first passage time. We show that the main result is particularly relevant to enhance our understanding of some problems of interest, such as the finite-time ruin probability of a dual compound Poisson risk model with diffusion and pricing barrier options issued on an insurer's stock price. Another closely related quantity of interest is the so-called occupation times of the surplus process below zero (also referred to as the duration of negative surplus, see, e.g., Egidio dos Reis (1993)) or in a certain interval (see, e.g., Kolkovska et al. (2005)). Occupation times have been widely used as a contingent characteristic to develop advanced derivatives in financial mathematics. In risk theory, it can be used as an important risk management tool to examine the overall health of an insurer's business. The main subject matter of Chapter 7 is to extend the analysis of occupation times to a class of renewal risk processes. We provide explicit expressions for the duration of negative surplus and the double-barrier occupation time in terms of their Laplace-Stieltjes transform. In the process, we revisit occupation times in the content of the classical compound Poisson risk model and examine some results proposed by Kolkovska et al. (2005). Finally, some concluding remarks and discussion of future research are made in Chapter 8.

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