Mathematical Theory Of Risk Research Articles

Ensuring the operational reliability of roads on the horizontal curves using TST-Risk software

Due to the urgency of the task to ensure the operational reliability of roads and compliance with the requirements of Technical Regulations of the Customs Union 014/2011 on road safety, we developed software to determine the danger of moving along the horizontal curve. The developed software is based on the office processing of experimental studies on the horizontal curve and mathematical risk theory models.

Development of a method to calculate the probability of a berth failure under vertical stochastic load

We developed the method for determining the probability of achieving the maximum value of loads from the cargo that is stored at the port terminal warehouse on the front wall of the under conditions of uncertainty of the moments of vessel approach and their loading. The method is based on the construction and analysis of the stochastic model of operation of port terminal, consisting of single and warehouse together with the reloading equipment. The process of the arrival of vessels with cargo is supposed to be described by the model of complex Poisson process; the whole cargo unloaded from the vessels arrives at warehouse and is transported from warehouse by the land transport in uniform way at constant intensity. The probabilistic model of the operation of the system a warehouse – berth at the port terminal was constructed with the help of the mathematical theory of risk. We formulated the criterion of safe operation of at its structural elements being influenced by the random load of the cargo, stored in warehouse in the rear of the berth, as the probability of not exceeding the maximum admissible value of the pressure on the front wall of the (according to the Coulomb law). To find the probability of failure, the linear integral equation of the convolution type, the solution to which was found for series of particular cases, was derived. Based on the found probability for failure, two practically important problems were stated and solved. First, the values of intensity of cargo transporting from warehouse, which ensures the absence of failure with the small enough assigned probability, were determined. Second, we stated the criterion of the economic viability of insuring the port losses as result of elimination of consequences of failure due to the maximum admissible value being exceeded by loads over the assigned period of time. The numerical illustration of the specified problems was presented

Chaos, Artificial Life and Risk Mathematics

The current work addresses the importance of chaos theory and artificial life as scientific bases for the development of a mathematical research of risk in the systems, within the framework of risk science and risk mathematics. It is argued, in particular, that a combination of chaos theory and artificial life may provide for a theoretical basis for the understanding of systemic situations of risk. It is presented, within the context of category computation theory, a notion of morphic incompressibility and it is addressed how statistical laws emerge in chaotic dynamics, which becomes morphically compressible at the statistical level, while being morphically incompressible at the individual orbits' level. The issue of the systemic origins of chaos is addressed in connection with this dual (orbit) incompressibility/(statistical) compressibility. Through the introduction of a one-dimensional game of life, that combines a coupled map lattice and a cellular automaton, in a model of an artificial ecosystem, it is, then, shown that artificial life can be effectively combined with chaos theory, in order to support the development of a mathematical theory of risk.

Upper bounds for ultimate ruin probabilities in the Sparre Andersen risk model with interest and a nonlinear dividend barrier

We consider the classical risk model with constant force of interest and a nonlinear dividend barrier. Lundberg-type inequalities for the ultimate ruin probabilities are derived. The results obtained carry over those of Gerber [Gerber, H.U., 1979. An Introduction to Mathematical Risk Theory. In: Monograph Series, vol. 8. Huebner Foundation, Philadelphia], about a linear dividend barrier without interest, to the case with both interest and a nonlinear dividend barrier. More precise upper bounds for the ultimate ruin probabilities are also given for the special case of exponential claim sizes.

An introduction to option pricing and the mathematical theory of risk

An introduction to option pricing and the mathematical theory of risk

Book Reviews

Abstract Abstract The Costs of Higher Education: How Much Do Colleges and Universities Spend Per Student and How Much Should They Spend?: Howard R. Buwen. San Francisco, Calif.: Jossey-Bass, 1980, xxiii + 287 pp. $15.95 (paperback). Reviewed by: Werner Z. Hirsch A Summary of Reports and Recommendations: The Carnegie Council on Policy Studies in Higher Education. San Francisco, Calif.: Jossey-Bass, 1980, x + 489 pp. $27.95 (paperback). Reviewed by: Charles S. Benson Sampling Methods for Censuses and Surveys (4th ed.): Frank Yates. New York: Macmillan, 1981. xvi + 458 pp. $55.00. Reviewed by: Charles H. Proctor Developments in Dual System Estimation of Population Size and Growth: Karol J. Krotki (ed.). Edmonton, Alberta, Canada: The University of Alberta Press, 1978. xviii + 260 pp. $17.50 (paperback). Reviewed by: Charles D. Cowan An Introduction to Mathematical Risk Theory: Hans U. Gerber. Philadelphia: S. S. Huebner Foundation for Insurance Education, 1979. xv + 164 pp. $15.95 (paperback). Reviewed by: E. Benton Cobb Mathematical Programming for Operations Researchers and Computer Scientists: Albert G. Holzman (ed.). New York: Marcel Dekker, 1981. xii + 373 pp. $45.00. Reviewed by: Patrick L. Brockett Techniques in Operational Research Vol. 2: Models, Search and Randomization.: Brian Conolly. New York: Ellis Horwood Limited Publishers (Division of John Wiley & Sons), 1981. 338 pp. $77.95. Reviewed by: Patrick L. Brockett Linear Programming: Algorithms and Applications.: S. Vajda. London, New York: Chapman and Hall, 1981. vii + 150 pp. $9.95 (paperback). Reviewed by: Salih O. Duffua Probabilistic Mechanical Design.: Edward B. Haugen. New York: John Wiley, 1980. 630 pp. $42.50. Reviewed by: Emil H. Jebe Engineering Applications of Correlation and Spectral Analysis.: Julius S. Bendat and Allan G. Piersol. New York: John Wiley, 1980. xiv + 302 pp. $29.95. Reviewed by: H. Dennis Tolley Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach.: Marcel F. Neuts. Baltimore, The Johns Hopkins University Press, 1981. xiii + 322 pp. $32.50. Reviewed by: C. C. Heyde Probability Inequalities in Multivariate Distributions.: Y. L. Tong. New York: Academic Press, 1980. xiii + 239 pp. $29.50. Reviewed by: Morris L. Eaton Applied Probability.: Frank A. Haight. New York: Plenum Press, 1981. xi + 290 pp. $35.00. Reviewed by: Luke Tierney Multinomial Probit: The Theory and Its Application to Demand Forecasting.: Carlos Daganzo. New York: Academic Press, 1979. xiv + 222 pp. $24.50. Reviewed by: B. B. Bhattacharyya Finite Mixture Distributions.: B. S. Everitt and D. J. Hand. New York: Methuen, 1981. xi + 143 pp. $14.95. Reviewed by: Stephen Vardeman Computational Probability.: P.M. Kahn (ed.). New York: Academic Press, 1980. xi + 340 pp. $21.00. Reviewed by: William F. Eddy Data Reduction: Analysing & Interpreting Statistical Data (revised reprint).: A.S.C. Ehrenberg. New York: John Wiley, 1978. xvii + 391 pp. Reviewed by: Brian L. Joiner Introductory Statistics for the Behavioral Sciences (4th ed.): Robert K. Young and Donald J. Veldman. New York: Holt, Rinehart and Winston, 1981. x + 687 pp. $19.95. Reviewed by: M. C. Shelley Basic Statistics for the Behavioral Sciences.: Kenneth Pfeiffer and James N. Olson. New York: Holt, Rinehart and Winston, 1981. xv + 444 pp. $19.95. Selltiz, Wrightsmαn and Cook's Research Methods in Social Relations (4th ed.): Louise H. Kidder. New York: Holt, Rinehart and Winston, 1981. xii + 483 pp. $14.95. Regression: A Second Course in Statistics.: Thomas H. Wonnacott and Ronald J. Wonnacott. New York: John Wiley, 1981. xix + 556 pp. $26.95. Reviewed by: Michael C. Costanza Intermediate Statistical Methods: G. Barrie Wetherill. London: Chapman and Hall, 1981. xv + 390 pp. $25.00. Reviewed by: Peter C. O'Brien Quantitive Methods for Decision Making in Business.: Richard E. Trueman. New York: Holt, Rinehart and Winston, 1981. 733 pp. Reviewed by: Anne B. Koehler Applied Statistical Techniques.: K.D.C. Stoodley, T. Lewis, and C.L.S. Stainton. New York: Halsted Press, 1980. 310 pp. $56.95. Reviewed by: George Thomas

Distribution Free Approximations in Applied Risk Theory

Most large insurance companies have today electronic computers that enable not only efficient actuarial statistics, but also research in applied risk theory. An important task in this latter field is the developing of an information system for control of the business as to the statistical balance between premiums and claims. The entire system can be separated into two parts, one descriptive and one analytic part. The descriptive part, that also may be called statistics production, is the base of the whole system and must be constructed in a general way to make it possible to apply mathematical tools in risk analysis. For the analytic part and its applications for computers there is a growing interest among actuaries as can be noticed from the reports in actuarial journals. The classical models of collective risk theory have recently been extensively illustrated by numerical calculations performed by the convolution committee in Sweden.When starting to construct the analytical part of the information system one finds that in spite of programming for the computer there is firstly a hard work to find realistic mathematical models, especially mathematical expressions for claim distributions, and secondly to estimate their parameters. According to the mathematical theory of risk, or the risk process, it is necessary to assume and test specific mathematical forms of two distributions, the number of claims and claims amount.

Reciprocal reinsurance treaties seen as a two-person co-operative game

Abstract 1.1. In 1930 Professor Cramer wrote: “The object of the theory of risk is to give a mathematical analysis of the random fluctuations in an insurance business, and to discuss the various means of protection against their inconvenient effects” ([6], page 7). This definition has obviously been adequate, since Cramer still subscribes to it when 25 years later he makes a comprehensive survey of the subject [7]. However, the mild complaint Cramer made in 1930 that “practical insurance business has hitherto made little or no application of the results offered by the mathematical theory of risk” still has some validity. The very impressive results achieved by the mathematical theory of risk during the last thirty years have certainly found practical applications, but not so many as one would expect, considering that the theory proposes to analyse the very foundations of the insurance business.

On the mathematical theory of risk

Although the first essay on the theory of risk (41) was published in 1786, this theory has been for the most part developed during this century. Moreover, the original presentation was rather difficult and the formulae inconvenient for the practical computer. Only recently has the classical theory of risk attained a satisfactory and seemingly final form. Few text-books, none of them in the English language, discuss these developments; it may therefore be useful to give an exposition of these results. As a rule no proofs will be given, instead reference will be made to the relevant literature.This survey will be restricted to the classical theory of risk, which deals essentially with the individual insurance. It is possible to approach the problem of risk from an entirely different angle, considering not the individual insurance but all the policies in force. This leads to the collective theory of risk (27, 29, 15, 40), which is based upon the theory of homogeneous random processes. Only the classical theory will be considered here, the inclusion of the collective theory of risk would considerably extend the scope of this paper, and in any case many of the relevant papers are in English.