# Stochastic Risk Analysis and Management von Harlamov, Boris (eBook)

• Erscheinungsdatum: 07.02.2017
• Verlag: Wiley-ISTE
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## Stochastic Risk Analysis and Management

The author investigates the Cramer -Lundberg model, collecting the most interesting theorems and methods, which estimate probability of default for a company of insurance business. These offer different kinds of approximate values for probability of default on the base of normal and diffusion approach and some special asymptotic.

## Produktinformationen

Format: ePUB
Seitenzahl: 164
Erscheinungsdatum: 07.02.2017
Sprache: Englisch
ISBN: 9781119388869
Verlag: Wiley-ISTE
Größe: 7362 kBytes
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## Stochastic Risk Analysis and Management

1
Mathematical Bases

1.1. Introduction to stochastic risk analysis

The concept of risk is diverse enough and is used in many areas of human activity. The object of interest in this book is the theory of collective risk. Swedish mathematicians Cramér and Lundberg established stochastic models of insurance based on this theory.

Stochastic risk analysis is a rather broad name for this volume. We will consider mathematical problems concerning the Cramér-Lundberg insurance model and some of its generalizations. The feature of this model is a random process, representing the dynamics of the capital of a company. These dynamics consists of alternations of slow accumulation (that may be not monotonous, but continuous) and fast waste with the characteristic of negative jumps.

All mathematical studies on the given subject continue to be relevant nowadays thanks to the absence of a compact analytical description of such a process. The stochastic analysis of risks which is the subject of interest has special aspects. For a long time, the most interesting problem within the framework of the considered model was ruin, which is understood as the capital of a company reaching a certain low level. Such problems are usually more difficult than those of the value of process at fixed times.

Let us consider the dynamics of the capital of an insurance company. It is supposed that the company serves several clients, which bring in insurance premiums, i.e. regular payments, filling up the cash desk of the insurance company. Insurance premiums are intended to compensate company losses resulting from single payments of great sums on claims of clients at unexpected incident times (the so-called insured events). They also compensate expenditures on maintenance, which are required for the normal operation of a company. The insurance company's activity is characterized by a random process which, as a rule, is not stationary. The company begins business with some initial capital. The majority of such undertakings come to ruin and only a few of them prosper. Usually they are the richest from the very beginning. Such statistical regularities can already be found in elementary mathematical models of dynamics of insurance capital.

The elementary mathematical model of dynamics of capital, the Cramér-Lundberg model, is constructed as follows. It uses a random process Rt ( t 0)
[1.1]
where u 0 is the initial capital of the company, p 0 is the growth rate of an insurance premium and pt is the insurance premium at time t . is a sequence of suit sizes which the insurance company must pay immediately. It is a sequence of independent and identically distributed (i.i.d.) positive random variables. We will denote a cumulative distribution function of U 1 (i.e. of all remaining) as B( x ) P (U1 x ) ( x 0). The function ( N t) ( t 0) is a homogeneous Poisson process, independent of the sequence of suit sizes, having time moments of discontinuity at points . Here, 0 s0 0.

Figure 1.1 shows the characteristics of the trajectories of the process.

Figure 1.1. Dynamics of capital

This is a homogeneous process with independent increments (hence, it is a homogeneous Markov process). Furthermore, we will assume that process trajectories are continuous from the right at any point of discontinuity.

Let 0 be a moment of ruin of the company. This means that at this moment, the company reaches into the neg

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