Ruin theory studies the fluctuations of these surplus processes. Introduction and main results originally motivated by pricing american claims, gerber and shiu 8, 9 introduced in risk theory a function. The book studies martingales and path decompositions, which are the main tools used in analysing the distribution of the time of ruin, the wealth prior to ruin. Shiu, this book gives a modern perspective on the problem of ruin for. Recent developments in exotic ruin theory are also considered. Optimal reinsurance for gerbershiu functions in the cram er. Shiu, this book gives a modern perspective on the problem of ruin for the classical cramerlundberg model and the surplus of. Bisand moralesgeneralized the gerbershiufunction topathdependentpenalties.
An algebraic approach to the analysis of gerbershiu functions. On the generalized gerbershiu function for surplus processes with. We assume that the risk free interest rate is constant through time and the stock pays no dividends. We propose a new estimation of gerber shiu function by an efficient method. This site is like a library, use search box in the widget to get ebook that you want. Garridoandmorales usedlaplacetransform toinvestigate theclassical gerber shiufunction. The title of these lecture notes may come as surprise to some readers as, to date, the term gerber. The gerbershiu discounted penalty functions for a risk model with. A note on a generalized gerbershiu discounted penalty. Some simulation examples are illustrated to show that the estimation has a good performance when the. The gerbershiu discounted penalty function in the risk. Gerbershiu analysis of a risk model with capital injections.
Introduction analyzing a risk model using the gerbershiu discounted function largely promoted the theory and provided a useful tool for the computation of many performance measures. We can study the laplace transform of the ruin time by the gerber shiu discounted penalty function, while other mathematical characteristics associated with the ruin time cannot be directly studied through gerber shiu discounted penalty function. In this paper, we consider the gerber shiu discounted penalty function for the sparre anderson risk process in which the interclaim times have a phase. See gerber and shiu 18 and schmidli 32 who deal with some special cases. Motivated by the many and longstanding contributions. On gerbershiu functions and optimal dividend distribution. May 15, 2010 ruin theory studies the fluctuations of these surplus processes. Classical problems include ruin low and dividend high barrier hitting times. Gerbershiu penalty function, markov modulated risk model, random income, delayed claims 1. It should be emphasized that most of the existing literatures mainly pursue explicit formulae of gerber shiu functions under various models. This paper shows that the esscher transform is also an efficient technique for valuing derivative securities if the logarithms of. Jun 28, 2016 we consider the risk model with capital injections studied by nie et al.
Weak approximations in risk theory have been used in iglehart 1969, grandell 1977, and recently in sarkar and sen 2005 to obtain the gerber shiu function for 3 in the case 2, and in furrer et al. Estimating the gerbershiu function under a risk model with. Over the decades, the importance of gerber shiu risk theory has advanced rapidly, attracting a massive number of research. Introduction analyzing a risk model using the gerber shiu discounted function largely promoted the theory and provided a useful tool for the computation of many performance measures.
Estimating the gerbershiu function in a compound poisson. In this paper, we study the gerbershiu distribution at parisian ruin for general l. Buy gerbershiu risk theory eaa series book online at best prices in india on. Each chapter corresponds to approximately two hours of lectures. Gerber and shiu 1998, a considerable amount of ruin theory research has been devoted to study this function. Over the decades, the importance of gerbershiu risk theory has advanced rapidly, attracting a massive number of research. We consider the risk model with capital injections studied by nie et al. The gerbershiu discounted penalty function for the bi. Shiu abstract the esscher transform is a timehonored tool in actuarial science. The purpose of this thesis is to provide an introduction to the basic concepts and common techniques used in the socalled gerber shiu analysis. Gerbershiu function in a class of delayed and perturbed. We construct a gerber shiu function and show that whilst this tool is not efficient for finding the ultimate ruin probability, it provides an effective way of studying ruin related quantities in finite time. Finally, in section we present somesimulation results toshow that.
Weak approximations in risk theory have been used in iglehart 1969, grandell 1977, and recently in sarkar and sen 2005 to obtain the gerbershiu function for 3 in the case 2, and in furrer et al. We construct a gerbershiu function and show that whilst this tool is not efficient for finding the ultimate ruin probability, it provides an effective way of studying ruin related quantities in finite time. On optimal dividend strategies in the compound poisson model. Fouriercosine method for gerbershiu functions sciencedirect. Originally motivated by pricing american claims, gerber and shiu 8, 9 introduced in risk theory a function that jointly penalizes the present value of the time of ruin, the surplus before ruin and. Gerber shiu function of markov modulated delayed byclaim. The biggest advancement to ruin theory after lundberg and cram er was achieved by powers in 1995 and gerber and shiu in 1998 who created the ex.
The situation in the nonhomogeneousdiscrete time risk model is much more complicated. If the discrete time risk model is generatedby possibly di. Two particular questions of interest in collective risk theory are a the deficit at ruin, and b the time of ruin, both of which have been treated separately in the literature. Incorporating these three values into a single function, gerbershiu function is used to assess the effect and the impact to an insurance company at the time of bankruptcy. By the compensation formula of the l evy process, the gerber shiu function admits an expression as a double integral with respect to the resolvent measure and the l evy measure. An algebraic operator approach to the analysis of gerber. The objective of this paper is to give an approximation to the gerber shiu function using the theory of scale functions. Click download or read online button to get risk theory book now. Optimal reinsurance for gerbershiu functions in the cram. Gerber shiu function is proposed by the observed sample of the surplus process. We can study the laplace transform of the ruin time by the gerbershiu discounted penalty function, while other mathematical characteristics associated with the ruin time cannot be directly studied through gerbershiu discounted penalty function. Bisand moralesgeneralized the gerber shiufunction topathdependentpenalties.
The gerber shiu expected penalty function for the risk model with dependence and a constant dividend barrier liu, donghai, liu, zaiming, and peng, dan, abstract and applied analysis, 2014. Risk theory for the compound poisson process that is perturbed by diffusion. In actuarial science and applied probability ruin theory sometimes risk theory collective risk theory uses mathematical models to describe an insurers vulnerability to insolvencyruin. Some simulation examples are illustrated to show that the estimation has a good performance when the sample size is. Shiu department of statistics and actuarial science the university of iowa iowa city, iowa 522421409, usa. Andreas kyprianou gerbershiu risk theory world of digitals. Gerbershiu risk theory can be used as lecture notes and is suitable for a graduate course. Reinsurance strategies are modelled as time dependant control functions, which leads to a setting from the theory of optimal stochastic control and ultimately to the problems hamiltonjacobibellman equation. The centre for actuarial studies at university of melbourne will host the 7th international gerbershiu workshop on 1011 july 2018 prior to the ime conference in sydney on 1618 july 2018. Andreas e kyprianou motivated by the many and longstanding contributions of h. However, my objective here is to focus on the recent interaction between a large body of research literature. Minca survey of matrix theory and matrix inequalities. In the last decade, the expected discounted penalty function, proposed by gerber and shiu 5, has unified the treatment of the joint distribution of the time to ruin, the surplus just prior to ruin.
Dynamic risk measures for stochastic asset processes from. Garridoandmorales usedlaplacetransform toinvestigate theclassical gerbershiufunction. Buy gerber shiu risk theory eaa series book online at best prices in india on. In this paper, we propose a new generalized gerbershiu discounted penalty function for a compound poisson risk model, which can be used to study the moments of the ruin time. This study proposes a novel mathematical justification of a risk measure for stochastic processes as a map on a functional path space of future loss. We propose a new estimation of gerber shiu function by an. Gerbershiu distribution at parisian ruin for l evy insurance. This note is an application of actuarial risk theory. In this way, the lundberg equation plays an important role in risk theory.
Gerber shiu risk theory can be used as lecture notes and is suitable for a graduate course. Estimating the gerbershiu expected discounted penalty. On the gerber shiu discounted penalty function in a risk model with two types of delayedclaims and random income. One might be more tempted to simply use the title ruin theory for cramerlundberg models. Incorporating these three values into a single function, gerber shiu function is used to assess the effect and the impact to an insurance company at the time of bankruptcy. Some dynamic risk measures, based on the expected discounted penalty function launched by gerber and shiu, are proposed to measure solvency risk from the companys goingconcern point of view. This paper shows that the esscher transform is also an efficient technique for valuing derivative securities if the logarithms of the prices of the prim. In particular, by making dividend or tax payments out of the surplus process, the effect on ruin is explored. Buy gerbershiu risk theory eaa series book online at. An algebraic operator approach to the analysis of gerbershiu. Gerber shiu function is proposed by the observed sample.
Gerber shiu penalty function, markov modulated risk model, random income, delayed claims 1. Gerbershiu distribution at parisian ruin for l evy. Then we can decompose the gerber shiu function as follows. Computing the gerbershiu function by frame duality. Hence the classical risk theory model is generalized by discounting with respect to the time of ruin. Buy gerbershiu risk theory eaa series book online at low. This function was rst studied in the classical poisson model where interclaim times are.
In this paper, we consider the gerbershiu discounted penalty function for the sparre anderson risk process in which the interclaim times have a phase. Baurdoux, juan carlos pardo, jose lius perez, jean. Gerbershiu function balbas 2010 major reference works. The themes of the workshop are risk theory, general insurance and related problems. Motivated by the many and longstanding contributions of h. Kyprianou1 department of mathematical sciences, university of bath 1basedonseveraljointworkswithe. Gerber shiu risk theory mathematical association of america. In this paper, we consider the behaviour of the special case of gerber shiu penalty. We show how to calculate an expected discounted penalty, which is due at ruin and may depend on the deficit at ruin and on the surplus immediately before ruin. We show that the estimator is easily computed and has a fast convergence rate. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin. In their model, ruin probabilities and the gerbershiu discounted penalty functions are studied, respectively. In insurance risk theory, the gerber shiu function introduced by gerber and shiu is a powerful tool for solving ruin problems.