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Elements of general risk theory

Prof. Vladimir Rykov
Russian State Oil&Gas University after Gubkin, Moscow

20 hours, 5 credits (final test)

September 15 - September 19, 2008
Dipartimento di Ingegneria dell'Informazione: Elettronica, Informatica, Telecomunicazioni, via G. Caruso, meeting room, ground floor

Contacts: Prof. Michele Pagano

Aims

During last years the term risk became very popular in different fields: economics, management, medicine, etc. and different authors use this term in different senses. It, may be, has many reasons, but this leads to the situation, that delays the development of this exclusively useful theory.

Historically among engineers the notion of risk has arisen in the framework of reliability theory and refers to the losses or damages for peoples, environment and equipment. From another side, among actuaries and mathematicians this term is used in the framework of ruin models. There is one more conception of risk, due to von Niemann and Morgenstern, which deals with the comparison of random variables in the framework of utility theory. Therefore, now there exist at least three different risk theories:

• engineering risk theory, which deals with investigation of the consequences (damages), generated by failures in complex non-reliable systems;
• insurance risk theory, which deals with the ruin problems;
• business risk theory, which considers the problems of random variables comparison.

In this situation a strong and common notion of risk, which could be used in inter-disciplinary connection, and the methods for the risk measurement and assessment are needed.

This series of lessons aims to give an introduction to the general risk theory, to propose the common for different applications notion of risk and to introduce different risk characteristics for different applications. The methodology of risk study in the framework of risk tree construction and analysis is also proposed.

Syllabus

0. Introduction (1 hour)

Part I. BACKGROUND

1. Risk notion (2 hours)

• Examples
• Notion of risk
• Classification of risk models

2. Risk measurement (2 hours)

• Risk distribution
• Risk intensity and hazard rates
• Generating functions and their properties
• Moments and other risk characteristics

3. Some classes of distributions and operations with them (2 hours)

• Some operations under r.v. and their distributions
• Heavy-tailed distributions
• Sub-exponential distributions

Part II. INDIVIDUAL RISK MODELS (ENGINEERING RISK THEORY)

4. Methodology for risk analysis (4 hours)

• Introducing remarks
• Preliminary risk analysis
• Risk tree construction
• Risk tree marking
• Risk tree analysis

5. Some parametric families of risk distributions and their characterization properties (2 hours)

• Some discrete parametric families of distributions
• Some parametric families of continuous distributions
• Distributions with monotone hazard rate

6. Compound distributions (2 hours)

• General properties
• Compound damages models
• Moments of compound damages

7. Structural reliability (2 hours)

• Notion of structural function
• Monotone structures
• Reliability of monotone system with independent elements
• Reliability function of monotone system

8. Qualitative risk analysis (2 hours)