Mathematical methods for studying risks (resumptive article)

We define risk as an unwanted opportunity and divide risk theory into three stages — risk analysis, risk estimation, risk management. Safety and risk are directly related to each other, being like a «mirror image» of each other which necessitates developing both the general theory of risk and particular theories of risk in specific areas. General risk theory allows for a uniform approach to the analysis, estimation and management of risks in specific situations. Currently, three main approaches to accounting for the uncertainty and describing risks are used — probabilistic and statistical approach, fuzzy sets, and the approach based on interval mathematics. The methods of risk estimation primarily based on probabilistic and statistical models are considered. The mathematical apparatus for estimating and managing risks is based on nonparametric formulations, limit relations, and multi-criteria optimization. Asymptotic nonparametric point estimates and confidence limits for the probability of a risk event are constructed on the base of binomial distribution and the Poisson distribution. Rules for testing statistical hypotheses regarding the equality (or difference) of two probabilities of risk events are proposed. An additive-multiplicative risk estimation model based on a hierarchical risk system based on a three-level risk system has become widespread: private risks — group risks — final risk. For this model, the role of expert estimation is revealed. The prospects of using (in the future) the theory of fuzzy sets are shown. The article deals with the main components of the mathematical apparatus of the theory of risks, in particular, the mathematical support of private theories of risks related to the quality management, innovations and investments. The simplest risk assessment in a probabilistic-statistical model is the product of the probability of a risk event and the mathematical expectation of the accidental damage. Mathematical and instrumental methods for studying global economic and environmental risks are discussed.

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