Articles

The paper studies the asymptotic behaviour of the mean total reserve of an insurance company in the case of a random number of clients. We use results on asymptotic deficiency and relative efficiency of procedures based on samples of fixed and random sizes. Zero-truncated Delaporte and generalized Poisson distributions are considered as models for the random number of clients. Asymptotic expansions of inverse moments are obtained for these distributions and the coefficients in the asymptotic formulas are computed. It is shown that the asymptotic deficiency of the scheme with a random number of clients can be expressed in terms of distribution parameters and in particular cases equals 1 and (1−θ)⁻². An empirical illustration is given based on data from the U.S. National Flood Insurance Program.

This work is devoted to studying methods for modeling and forecasting the value of underlying assets based on market option prices. The use of options allows one to recover probability distributions of future asset prices and construct their volatility surfaces, which is necessary for modeling asset dynamics and valuing derivative financial instruments. Non-parametric approaches based on the extraction of risk-neutral probability density functions from option prices are considered, as well as parametric implied volatility surface models, such as SVI and SABR, which provide input for constructing the dynamic Dupire local volatility model. Dynamic models of asset prices with calibration based on market option quotes are also discussed, including the Heston model and discrete ARMA-GARCH models.

In the article, the asymptotics of the coefficients of the generating functions are found with a certain accuracy, which can be used to calculate the powers of layers of certain types of partially ordered sets, as well as to calculate the values of the sums of boundary functionals when estimating the number of antichains in such sets. In addition, applications of the obtained results are considered using examples.

A nonlinear two-phase mathematical model of production at a mining plant is proposed, where ore flow is represented as two phases connected to each other through concentration of useful substance contained in the ore. The process of material balancing is studied, which leads to optimization problem with nonlinear constraints. The interior point method is used to solve the problem. Known measured values of useful substance concentrations are taken into account, as well as the results of indirect measurements, ranges of flow values, unknown concentration values, limits on stored resources. Test experiments are performed, and calculation results are demonstrated.

Initial boundary-value problem for the evolutional equation with involution of the space variable and a non-local boundary condition is studied. The uniqueness of the problem’s solution is obtained and classes of initial functions which provide solution’s existence and stability are described.

A certain characterization of the logistic distribution is proposed by the property of identical distribution of two linear forms.

This work addresses the problem of single-variable function approximation within the class of nonlinear approximations with two sets of parameters. The approximating function is linear with respect to the first parameter (which is constrained to be positive), while each term of the approximating function represents a given function with nonlinear dependence on the second parameter. The computational algorithm is based on residual minimization in Hilbert space. The first key aspect of the developed approach involves effectively reducing the problem to a standard linear best approximation problem by discretizing the second parameter over an extended set of points within the admissible segment. The second key feature lies in determining the set of linear approximation parameters at each iteration of the classical non-negative least squares (NNLS) method. Numerical results are presented to demonstrate the capabilities of this computational algorithm for nonlinear function approximation.

This work examines a modification of a spatially distributed Lanchester model, which describes an antagonistic game between two groups in a two-dimensional domain. The model originally emerged in 1916 to describe combat scenarios between two military forces, characterizing events of the First World War. The model accounts for directed strikes, defined by a velocity vector, and the initial displacement of troop concentrations using a step function. The system includes diffusion terms, nonlinear reactions, Gaussian white noise, and a topographic obstacle (a lake of arbitrary shape) that restricts troop movement. The numerical solution is implemented using the finite element method based on domain triangulation to handle irregular domains. The simulation results demonstrate an increase in spatial heterogeneity under the influence of noise and the obstacle, as well as a significant impact of directed movement and initial displacement on conflict dynamics. Stability analysis confirms the system’s stability, including an analysis specific to the finite element method. Sensitivity analysis of the parameters and computational error estimation have been added. Visualization illustrates the evolution of troop densities over time. The advantages of the proposed approach over existing methods include better handling of irregular domains and model scalability.

The paper considers the asymptotic behaviour of the distribution functions of the statistics based on the samples with random sizes in the testing problems conserning univariate parameters. An asymptotic comparison of the tests is carried out in terms of the necessary additional number of such factors (asymptotic deficiency). Two examples illustrating the obtained results are presented. These examples concern truncated Poisson and binomial distributions are considered.

The paper considers a method of stabilized hard thresholding in the problem of inverting linear homogeneous operators using wavelet decomposition. In a data model with additive Gaussian noise, an analysis of the unbiased estimate of the mean square risk of this method is carried out. Under the assumption of a long-term dependence between noise coefficients, conditions are given under which the unbiased risk estimate is strongly consistent and asymptotically normal.

The problem of Hausdorff approximation by finite sets of the solution and the value of a multicriteria mixed strategy bimatrix game using a representation based on linear scalarization is considered. For the case of 2×2 matrices, explicit formulas are obtained for constructing nodes of a δ-net on the product of simplices of scalarization parameters. It is proved that the set of the equilibrium values obtained for the net converges in the Hausdorff metric to the solution of the initial game at δ →0. Possibile appearance of degenerate bimatrix games in scalarization is taken account. Examples are given for two-criteria 2×2×2 games.

In this paper, the definitions of generalized Student’s distributions are extended to a wider set of parameters of these distributions and multiplication theorems are given that allow the generalized Student’s and Lomax’s distributions to be represented as scale mixtures of the same distributions but with larger parameters. A similar result is obtained for beta distributions. Analogs of multiplication theorems are obtained for the classical Student’s and Lomax’s distributions as corollaries; in particular, it is shown that the Student’s distribution can be represented as a scale mixture of the Student’s distribution with a large number of degrees of freedom. A representation of strictly stable distributions concentrated on the positive semiaxis is also obtained as scale mixtures of a special distribution that is not stable. This alternative representation complements the multiplication theorem for such strictly stable laws.

We present an efficient algorithm for checking language equivalence of states in top-down deterministic finite tree automata (DFTAs). Unlike string automata, tree automata operate over hierarchical structures, posing unique challenges for algorithmic analysis. Our approach reduces the equivalence checking problem to that of checking the solvability of a system of language-theoretic equations which specify the behavior of a DFTA. By constructing such a system of equations and systematically manipulating with it through substitution and conflict detection rules, we develop a decision procedure that determines whether two states accept the same tree language. We formally prove the correctness and termination of the algorithm and establish its worst-case time complexity as O(n²) under the RAM (Random Access Machine) model of computation augmented with pointers.

The aim of equivalence testing is to verify that two parameters are sufficiently close or, alternatively, that the parameter of interest lies within two pre-specified limits. The two one-sided tests procedure is arguably the most widely known approach for assessing equivalence in the pharmaceutical field. Using a model that accounts for missing data, it is shown analytically that the type I error rate may exceed the nominal significance level. A refined estimate of this error is also obtained. For the standard 2x2 crossover design, a method is proposed that enables control of the type I error in the presence of missing data.