The work is devoted to the development of methods for dynamic risk measures VaR and CVaR estimating. As a basic model, a heteroscedastic time series model is considered. The methods proposed in the article are designed for obtaining the forecast estimates of risk measures for volatile time series taking into account the long-range dependence presence. The method of smoothing of the autocorrelation function based on an optimization procedure is used for variance modeling. A metalog distribution is proposed to use for risk measures model residuals estimating. This distribution allows to describe the behavior of the tail part of the distribution with different characteristics. The paper proposes two methods of metalog distribution estimating. The first method is based on an empirical distribution function and the second one on its approximation by sample quantiles. For VaR and CVaR modeling and forecasting, explicit analytical formulas were obtained with different numbers of members of the metalog distribution. The procedure for obtaining of the forecast values of dynamic risk measures VaR and CVaR is formulated as an algorithm. The proposed approach is applied to the time series of the "Russian Trading System" index for the period 14/10/2005 -10/02/2020. For comparison, the forecast of dynamic risk measures is built using well known methods of risk estimation based on the GEV distribution, GPD and historical modeling. Quantitative and qualitative analyzes of the obtained estimates confirmed the high quality of the obtained estimates.

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This work focuses on development of a mathematical apparatus that allows to perform an approximate description of inhomogeneities of finite sizes in a continuous bodies by arranging the sources given on sets of smaller dimensions. The structure and properties of source densities determine the adequacy of the model. The theory of differential forms and generalized functions underlies this study. The boundary value problems with nonsmooth coefficients are formulated. The solutions of such problems is sought in the form of weakly convergent series and as an alternative - an equivalent recurrent set of boundary value problems with jumps. A feature of this approach is the ability to consistently improve the adequacy of the description of inhomogeneity. This is important because it allows to qualitatively assess the impact of real characteristic properties on the accuracy of the model description. Reducing the dimensions of inhomogeneities allows the use of efficient methods such as the Green's function and boundary integral equations to obtain a semi-analytic solution for direct and inverse problems. The work is based on a number of partial problems that demonstrate the proposed approach in modeling of inhomogeneities. The problems of modeling of the set of finite defects in an oscillating elastic beam, the set of inhomogeneities of an arbitrary shape in an oscillating plate, fragile cracks in a two-dimensional elastic body under static loading are considered.

The paper proposes a new approach to the construction of point defect models, based on the solution of boundary value problems with non smooth coefficients. Heterogeneity is included in the determining equation of the boundary problem. This approach allows us to formalize defects at the stage of use of state equations, and thus automatically reconciles the defect with the hypotheses of diminution of dimension and does not break the energy closed. The solution is sought in the form of weakly convergent series of generalized functions. The proposed approach simplifies the mechanical interpretation of defect parameters and is demonstrated in several examples. In the first example, the Green function for harmonic oscillations of an elastic beam with a point defect is constructed. The defect model is a limiting state of elastic inclusion with weakening or strengthening. The second example considers the inclusion of an elliptical shape in the problem of harmonic oscillations of the elastic plate. The first approximation of the equivalent volumetric force is constructed and the path to the following approximations is indicated. In the third example, a model of a brittle crack with a known displacement jump is constructed for a static two-dimensional problem of elasticity theory.

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