30 research outputs found
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
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 distribution of the time to ruin and related topics
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
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
Gerber–Shiu function in a class of delayed and perturbed risk model with dependence
Abstract:This paper considers the risk model perturbed by a diffusion process with a time delay in the arrival of the first two claims and takes into account dependence between claim amounts and the claim inter-occurrence times. Assuming that the time arrival of the first claim follows a generalized mixed equilibrium distribution, we derive the integro-differential Equations of the Gerber–Shiu function and its defective renewal equations. For the situation where claim amounts follow exponential distribution, we provide an explicit expression of the Gerber–Shiu function. Numerical examples are provided to illustrate the ruin probability
On modeling quantities for insurer solvency against catastrophe under some Markovian assumptions
Insurance companies sometimes face catastrophic losses, yet they must remain solvent enough to meet the legal obligation of covering all claims. Catastrophes can result in large damages to the policyholders, causing the arrival of numerous claims to insurance companies at once. Furthermore, the severity of an event could impact the time until the next occurrence. An insurer needs certain levels of startup capital to meet all claims, and then must have adequate reserves on a continual basis, even more so when catastrophes occur. This work examines two facets of these matters: for an infinite time horizon, we extend and develop models for insurer bankruptcy-related quantities accounting for the reality of large claims occurring. Meanwhile, for finite time horizons, we model the present value of claims that have been incurred but not yet reported, so-called \u27IBNR\u27 claims. In the former, we show how our method for \u27Gerber-Shiu\u27 functions works in a recently proposed dependency structure allowing insurers to charge clients different premiums depending on their riskiness. In the latter, we build upon a recent method which allowed claims to arrive in batches; besides permitting discounting to be time-dependent, we allow the insurer to adjust the assumed distribution of the time until the next event by comparing the number of claims from the current event to any number of random intervals. We provide numerical studies for both scenarios --Abstract, page iii
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