Self-similar models in risk theory

This Ph.D. thesis is concerned with self-similar processes. In Chapter 2 we describe the classes of transformations leading from self-similar to stationary processes, and conversely. The relationship is used in Chapter 3 to characterize stable symmetric self-similar processes via their minimal integral representation. This leads to a unique decomposition of a symmetric stable self-similar process into three independent parts. The class of such processes appears to be quite broad and can stand as a basis of different risk models. In Chapter 4 we give examples of applications of self-similar processes in insurance risk modelling. In Chapter 5 we illustrate a test of self-similarity (namely variance-time plots) on DJIA index data in order to justify the use of self-similar processes in financial modelling. Last but not least we propose an alternative model for stock price movements incorporating a martingale which generates the same filtration as fractional Brownian motion.Self-similar process; Risk theory; Lamperti transformation; Insurance; Option pricing;

Modeling the risk process in the XploRe computing environment

A user friendly approach to modeling the risk process is presented. It utilizes the insurance library of the XploRe computing environment which is accompanied by on-line, hyperlinked and freely downloadable from the web manuals and e-books. The empirical analysis for Danish fire losses for the years 1980-90 is conducted and the best fitting of the risk process to the data is illustrated.Risk process, Monte Carlo simulation, XploRe computing environment

Modeling the risk process in the XploRe computing environment

A user friendly approach to modeling the risk process is presented. It utilizes the insurance library of the XploRe computing environment which is accompanied by on-line, hyperlinked and freely downloadable from the web manuals and e-books. The empirical analysis for Danish fire losses for the years 1980-90 is conducted and the best fitting of the risk process to the data is illustrated. --

Property insurance loss distributions

Property Claim Services (PCS) provides indices for losses resulting from catastrophic events in the US. In this paper we study these indices and take a closer look at distributions underlying insurance claims. Surprisingly, the lognormal distribution seems to give a better fit than the Paretian one. Moreover, lagged autocorrelation study reveals a mean-reverting structure of indices returns.Econophysics; Property insurance; Loss distribution; PCS index;

Equity-linked insurances and guaranteed annuity options

We consider here term and whole-life cases of the equity-linked life insurance(ELLI), and the guaranteed annuity option (GAO). We present a financial instrument which is a combination of ELLI and GAO in a stochastic interest rate framework.equity-linked life insurance; guaranteed annuity option; geometric Brownian motion; Ornstein-Uhlenbeck process; Monte Carlo simulations

Building Loss Models

This paper is intended as a guide to building insurance risk (loss) models. A typical model for insurance risk, the so-called collective risk model, treats the aggregate loss as having a compound distribution with two main components: one characterizing the arrival of claims and another describing the severity (or size) of loss resulting from the occurrence of a claim. In this paper we first present efficient simulation algorithms for several classes of claim arrival processes. Then we review a collection of loss distributions and present methods that can be used to assess the goodness-of-fit of the claim size distribution. The collective risk model is often used in health insurance and in general insurance, whenever the main risk components are the number of insurance claims and the amount of the claims. It can also be used for modeling other non-insurance product risks, such as credit and operational risk.Insurance risk model; Loss distribution; Claim arrival process; Poisson process; Renewal process; Random variable generation; Goodness-of-fit testing

An introduction to simulation of risk processes

A typical model for insurance risk, the so-called collective risk model, has two main components: one characterizing the frequency (or incidence) of events and another describing the severity (or size or amount) of gain or loss resulting from the occurrence of an event. Here we focus on simulating the point process N(t) of the incidence of events. We discuss five prominent examples of N(t), namely the classical (homogeneous) Poisson process, the non-homogeneous Poisson process, the mixed Poisson process, the Cox process (also called the doubly stochastic Poisson process) and the renewal process.Collective risk model; Poisson process; Non-homogeneous Poisson process; Mixed Poisson process; Cox process; Renewal process;

The Lamperti transformation for self-similar processes

In this paper we establish the uniqueness of the Lamperti transformation leading from self-similar to stationary processes, and conversely. We discuss alpha-stable processes, which allow to understand better the difference between the Gaussian and non-Gaussian cases. As a by-product we get a natural construction of two distinct alpha-stable Ornstein–Uhlenbeck processes via the Lamperti transformation for 0Lamperti transformation; Self-similar process; Stationary process; Stable distribution;

Building Loss Models

This paper is intended as a guide to building insurance risk (loss) models. A typical model for insurance risk, the so-called collective risk model, treats the aggregate loss as having a compound distribution with two main components: one characterizing the arrival of claims and another describing the severity (or size) of loss resulting from the occurrence of a claim. In this paper we first present efficient simulation algorithms for several classes of claim arrival processes. Then we review a collection of loss distributions and present methods that can be used to assess the goodness-of-fit of the claim size distribution. The collective risk model is often used in health insurance and in general insurance, whenever the main risk components are the number of insurance claims and the amount of the claims. It can also be used for modeling other non-insurance product risks, such as credit and operational risk.Insurance risk model; Loss distribution; Claim arrival process; Poisson process; Renewal process; Random variable generation; Goodness-of-fit testing;