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.

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.

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.

The aim of this thesis is to develop algorithmic methods for computing particular performance measures of interest for a general class of discrete-based insurance risk models. We build upon and generalize the insurance risk models considered by Drekic and Mera (2011) and Alfa and Drekic (2007), by incorporating a threshold-based dividend system in which dividends only get paid provided some period of good financial health is sustained above a pre-specified threshold level. We employ two fundamental methods for calculating the performance measures under the more general framework. The first method adopts the matrix-analytic approach originally used by Alfa and Drekic (2007) to calculate various ruin-related probabilities of interest such as the trivariate distribution of the time of ruin, the surplus prior to ruin, and the deficit at ruin. Specifically, we begin by introducing a particular trivariate Markov process and then expressing its transition probability matrix in a block-matrix form. From this characterization, we next identify an initial probability vector for the process, from which certain important conditional probability vectors are defined. For these vectors to be computed efficiently, we derive recursive expressions for each of them. Subsequently, using these probability vectors, we derive expressions which enable the calculation of conditional ruin probabilities and, from which, their unconditional counterparts naturally follow. The second method used involves the first claim conditioning approach (i.e., condition on knowing the time the first claim occurs and its size) employed in many ruin theoretic articles including Drekic and Mera (2011). We derive expressions for the finite-ruin time based Gerber-Shiu function as well as the moments of the total dividends paid by a finite time horizon or before ruin occurs, whichever happens first. It turns out that both functions can be expressed in elegant, albeit long, recursive formulas. With the algorithmic derivations obtained from the two fundamental methods, we next focus on computational aspects of the model class by comparing six different types of models belonging to this class and providing numerical calculations for several parametric examples, highlighting the robustness and versatility of our model class. Finally, we identify several potential areas for future research and possible ways to optimize numerical calculations.

The aim of this thesis is to develop algorithmic methods for computing particular performance measures of interest for a general class of discrete-based insurance risk models. We build upon and generalize the insurance risk models considered by Drekic and Mera (2011) and Alfa and Drekic (2007), by incorporating a threshold-based dividend system in which dividends only get paid provided some period of good financial health is sustained above a pre-specified threshold level. We employ two fundamental methods for calculating the performance measures under the more general framework. The first method adopts the matrix-analytic approach originally used by Alfa and Drekic (2007) to calculate various ruin-related probabilities of interest such as the trivariate distribution of the time of ruin, the surplus prior to ruin, and the deficit at ruin. Specifically, we begin by introducing a particular trivariate Markov process and then expressing its transition probability matrix in a block-matrix form. From this characterization, we next identify an initial probability vector for the process, from which certain important conditional probability vectors are defined. For these vectors to be computed efficiently, we derive recursive expressions for each of them. Subsequently, using these probability vectors, we derive expressions which enable the calculation of conditional ruin probabilities and, from which, their unconditional counterparts naturally follow. The second method used involves the first claim conditioning approach (i.e., condition on knowing the time the first claim occurs and its size) employed in many ruin theoretic articles including Drekic and Mera (2011). We derive expressions for the finite-ruin time based Gerber-Shiu function as well as the moments of the total dividends paid by a finite time horizon or before ruin occurs, whichever happens first. It turns out that both functions can be expressed in elegant, albeit long, recursive formulas. With the algorithmic derivations obtained from the two fundamental methods, we next focus on computational aspects of the model class by comparing six different types of models belonging to this class and providing numerical calculations for several parametric examples, highlighting the robustness and versatility of our model class. Finally, we identify several potential areas for future research and possible ways to optimize numerical calculations.

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas.

As ruin theory evolves in recent years, there has been a variety of quantities pertaining to an insurer's bankruptcy at the centre of focus in the literature. Despite the fact that these quantities are distinct from each other, it was brought to our attention that many solution methods apply to nearly all ruin-related quantities. Such a peculiar similarity among their solution methods inspired us to search for a general form that reconciles those seemingly different ruin-related quantities. The stochastic approach proposed in the thesis addresses such issues and contributes to the current literature in three major directions. (1) It provides a new function that unifies many existing ruin-related quantities and that produces more new quantities of potential use in both practice and academia. (2) It applies generally to a vast majority of risk processes and permits the consideration of combined effects of investment strategies, policy modifications, etc, which were either impossible or difficult tasks using traditional approaches. (3) It gives a shortcut to the derivation of intermediate solution equations. In addition to the efficiency, the new approach also leads to a standardized procedure to cope with various situations. The thesis covers a wide range of ruin-related and financial topics while developing the unifying stochastic approach. Not only does it attempt to provide insights into the unification of quantities in ruin theory, the thesis also seeks to extend its applications in other related areas.