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## Über dieses Buch

Motivated by the many and long-standing contributions of H. Gerber and E. Shiu, this book gives a modern perspective on the problem of ruin for the classical Cramér–Lundberg model and the surplus of an insurance company. 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 and the deficit at ruin. Recent developments in exotic ruin theory are also considered. In particular, by making dividend or tax payments out of the surplus process, the effect on ruin is explored.

Gerber-Shiu Risk Theory can be used as lecture notes and is suitable for a graduate course. Each chapter corresponds to approximately two hours of lectures.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
In this brief introductory chapter, we outline the basic context of these lecture notes. In particular, we explain what we understand by so-called Gerber–Shiu theory and the role that it has played in classical ruin theory.
Andreas E. Kyprianou

### Chapter 2. The Wald Martingale and the Maximum

Abstract
We introduce the first of our two key martingales and consider two immediate applications. In the first application, we will use the martingale to construct a change of measure and thereby consider the dynamics of X under the new law. In the second application, we shall use the martingale to study the law of the process $$\overline{X}= \{\overline{X}_{t}\,\colon t\geq 0\}$$, where
$$\overline{X}_t =\sup_{s\leq t}X_s, \quad t\geq 0.$$
In particular, we shall discover that the position of the trajectory of $$\overline{X}$$, when sampled at an independent and exponentially distributed time, is again exponentially distributed.
Andreas E. Kyprianou

### Chapter 3. The Kella–Whitt Martingale and the Minimum

Abstract
We move to the second of our two key martingales. In a similar spirit to Chapter 2, we shall use the martingale to study the law of the process $$\underline{X}= \{\underline{X}_{t}\,\colon t\geq 0\}$$, where
$$\underline{X}_t : =\inf_{s\leq t}X_s, \quad t\geq 0.$$
As with the case of $$\overline{X}$$, we characterise the law of $$\underline{X}$$ when sampled at an independent and exponentially distributed time. Unlike the case of $$\overline{X}$$ however, this will not turn out be exponentially distributed.
Andreas E. Kyprianou

### Chapter 4. Scale Functions and Ruin Probabilities

Abstract
The two main results from the previous chapters concerning the law of the maximum and minimum of the Cramér–Lundberg process can now be put to use in order to establish our first results concerning the classical ruin problem. We introduce the so-called scale functions, which will prove to be indispensable, both in this chapter and later, when describing various distributional features of the ruin problem.
Andreas E. Kyprianou

### Chapter 5. The Gerber–Shiu Measure

Abstract
Having introduced scale functions, we look at the Gerber–Shiu measure, which characterises the joint law of the discounted wealth prior to ruin and deficit at ruin. We shall develop an idea from Chapter 4, involving Bernoulli trials of excursions from the minimum, to provide an identity for the expected occupation measure until ruin of the Cramér–Lundberg process. This identity will then play a key role in identifying an expression for the Gerber–Shiu measure.
Andreas E. Kyprianou

### Chapter 6. Reflection Strategies

Abstract
We to the first of the three cases in which the path of the Cramér–Lundberg process is perturbed through the payments of dividends. Recall that a reflection (or barrier) strategy consists of paying dividends out of the surplus in such a way that, for a fixed barrier a>0, any excess of the surplus above this level is instantaneously paid out. The key object of interest in this chapter is the present value of the dividends paid until ruin under force of interest.
Andreas E. Kyprianou

### Chapter 7. Perturbation-at-Maximum Strategies

Abstract
In this chapter, we analyse the case in which payments are made from the surplus that are proportional to increments of the maximum. With appropriate assumptions, this perturbation to the path of the surplus may be seen as the result of taxation.
Andreas E. Kyprianou

### Chapter 8. Refraction Strategies

Abstract
We consider the refracted Cramér–Lundberg process, that is, the solution to the stochastic differential equation (SDE)
$$\mathrm {d}Z_t = \mathrm {d}X_t - \delta \mathbf{1}_{(Z_t >\mathrm{b})}\,\mathrm {d}t, \quad t\geq 0,$$
for some threshold b≥0. We charge ourselves with the task of providing identities for the probability of ruin as well as the expected present value of dividends paid until ruin. It turns out that all identities can be written in terms of the scale functions of two different processes. One scale function comes from the Cramér–Lundberg process X and the other from the same Cramér–Lundberg process but with premium rate reduced by δ.
Andreas E. Kyprianou

### Chapter 9. Concluding Discussion

Abstract
On the one hand, the use of scale functions would appear to have made many of the problems that we have considered at in previous chapters look solvable. On the other hand, one may question the extent to which we have solved the posed problems, as our scale functions are only defined in terms of a Laplace transform. We have arguably only provided a solution “up to the inversion of a Laplace transform”. It would be nice to have some concrete examples of scale functions. It turns out that few concrete examples are known and they are quite difficult to produce in general. Nonetheless, we shall show that there is still sufficient analytical structure known for a general scale function to justify their use, in particular when moving to the bigger class of processes for which the surplus process is modelled by a general spectrally negative Lévy process.
Andreas E. Kyprianou

### Backmatter

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