Stochastic Optimization in Insurance

A Dynamic Programming Approach

verfasst von: Pablo Azcue, Nora Muler

Verlag: Springer New York

Buchreihe : SpringerBriefs in Quantitative Finance

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The main purpose of the book is to show how a viscosity approach can be used to tackle control problems in insurance. The problems covered are the maximization of survival probability as well as the maximization of dividends in the classical collective risk model. The authors consider the possibility of controlling the risk process by reinsurance as well as by investments. They show that optimal value functions are characterized as either the unique or the smallest viscosity solution of the associated Hamilton-Jacobi-Bellman equation; they also study the structure of the optimal strategies and show how to find them.

The viscosity approach was widely used in control problems related to mathematical finance but until quite recently it was not used to solve control problems related to actuarial mathematical science. This book is designed to familiarize the reader on how to use this approach. The intended audience is graduate students as well as researchers in this area.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Stability Criteria for Insurance Companies

Abstract

In this chapter we present the classical collective risk model for an insurance company and introduce two ways of measuring the stability of the company: survival probability and the maximization of the expectation of the discounted dividend payments. We consider these stability measures as functions of the initial surplus; they are called the value functions of the corresponding problems. We present here the bare case; in later chapters we will also allow the company to control the risk by means of reinsurance and investment.

Pablo Azcue, Nora Muler

Chapter 2. Reinsurance and Investment

Abstract

In this chapter we present the two main ways to control the insurance risk process: reinsurance and investment. We focus on the classical risk model.

Pablo Azcue, Nora Muler

Chapter 3. Viscosity Solutions

Abstract

In Chaps. 1 and 2 we have obtained heuristically the associated equations to the value functions of different control problems. We cannot expect in general to have optimal value functions smooth enough to satisfy these equations in the classical sense. In this chapter we explain the notion of viscosity solutions for ordinary integrodifferential equations and show that the optimal value functions for the classical risk model are indeed solutions of the associated equations in this weaker sense.

Pablo Azcue, Nora Muler

Chapter 4. Characterization of Value Functions

Abstract

This chapter is devoted to characterize the optimal value functions among the viscosity solutions of the corresponding HJB equations in the classical risk model. We consider the bare case presented in (1.6) and (1.10), the case with reinsurance presented in (2.9) and (2.16), and the case with investment presented in (2.28) and(2.33).

Pablo Azcue, Nora Muler

Chapter 5. Optimal Strategies

Abstract

The aim of the present chapter is to show the existence of optimal stationary strategies in the classical risk models. We start with the problems of dividend payments and consider first the simplest problem without reinsurance or investment control.

Pablo Azcue, Nora Muler

Chapter 6. Numerical Examples

Abstract

In this chapter we show some examples of the optimal value functions and the optimal strategies for the classical risk model. In these examples, the optimal band strategies have one (barrier) or two bands; we have not found examples with more bands in the unbounded dividend payment case. However, when imposing a ceiling on the rate of dividends, band strategies with infinitely many bands can be found (even with claim-size distributions with bounded density); see [12].

Pablo Azcue, Nora Muler

Backmatter

Titel: Stochastic Optimization in Insurance
verfasst von: Pablo Azcue
Nora Muler
Verlag: Springer New York
Electronic ISBN: 978-1-4939-0995-7
Print ISBN: 978-1-4939-0994-0
DOI: https://doi.org/10.1007/978-1-4939-0995-7