Modeling of risk in multidimensional stochastic systems

In this paper a new approach to modeling and research of risk of multidimensional stochastic systems of various natures is offered. The hypothesis that the risk can be operated due to change of probabilistic properties a component of multidimensional stochastic system is the cornerstone of the offered risk model. At the same time the multidimensional stochastic system is modelled in the form of a random vector with components generally mutually correlated. The problem of risk minimization for multidimensional stochastic system is formulated. The special case of risk minimization of Gaussian stochastic systems where the operating variables are numerical characteristics of a random vector (a covariance matrix and a vector of expectations) is considered. For arbitrary distributions of the random vector components the approximate algorithms of decrease the risk based on use of the multidimensional statistical analysis methods are proposed. The cluster analysis and the multidimensional regression analysis can be carried to these methods. For Gaussian random vectors calculations of risk for cases uncorrelated and correlated components are given. Now at research of complex multidimensional systems risk, they do not allocate in an explicitly their components. As showed modeling, unaccounted in an explicit form multidimensionality of system and mutual correlation of its components can lead to significant underestimation of the actual risk. The paper gives examples of calculations for multidimensional Gaussian random vectors. Dimensions were taken from 1 to 5. The results of calculation of probability of dangerous outcome depending on numerical characteristics of a multidimensional Gaussian random variable (a covariance matrix and a vector of expectations) are given. Different versions of areas of dangerous outcomes are considered. For correct application of the described risk model for the composite stochastic systems, it is necessary to use as its components essential factors which objective reflect the cause and effect regularities proceeding in these systems.

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