The method of successive approximation applied to find the probability for an insurance company with random premiums

A risk process that describes the evolution of the capital of an insurance company is analyzed, random premiums and claims being available. Integral equations of nonbankruptcy probability as a function of the initial capital are derived. Necessary and sufficient conditions for the existence and uniqueness of solutions of these integral equations, and convergence conditions for the method of successive approximation for finding their solutions are established __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 112–127, January–February 2006.     

Asymptotic efficiency of kernel support vector machines (SVM)

The paper analyzes the asymptotic properties of Vapnik’s SVM-estimates of a regression function as the size of the training sample tends to infinity. The estimation problem is considered as infinite-dimensional minimization of a regularized empirical risk functional in a reproducing kernel Hilbert space. The rate of convergence of the risk functional on SVM-estimates to its minimum value is established. The sufficient conditions for the uniform convergence of SVM-estimates to a true regression function with unit probability are given. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 81–97, July–August 2009     

Solving the Wiener-Hopf equation with a probabilistic kernel

Inhomogeneous Wiener-Hopf equations on a halfline with probabilistic kernels are considered. They are analyzed using the theory of isotonic and contraction operators. Necessary and sufficient conditions of existence and general sufficient conditions of uniqueness for the solutions of these equations are established. The conditions for and the rate of convergence of successive approximations of the solution are found. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 39–47, March–April 2006.     

Equivalence of the probability measures generated by the solutions of nonlinear evolution differential equations in a Hilbert space,disturbed by Gaussian processes. Part I

Nonlinear evolution differential equations with unbounded linear operators of disturbance by Gaussian random processes are considered in an abstract Hilbert space. For the Cauchy problem for the differential equations, the sufficient existence and uniqueness conditions for their solutions are proved and the sufficient conditions for the equivalence of the probability measures generated by these solutions are derived. Moreover, the corresponding Radon–Nikodym densities are calculated explicitly in terms of the coefficients or characteristics of the considered differential equations.     

An optimal control model for a system of degenerate parabolic integro-differential equations

An initial-boundary-value problem for a system of degenerate parabolic integro-differential equations is considered. The sufficient conditions for the existence and uniqueness of its generalized solution and for the existence of at least one optimal control for a given performance functional are obtained. A stable numerical solution to the initial-boundary-value problem is derived for a locally one-dimensional case and conditions are formulated for constructing a stable numerical algorithm of the optimal control problem on a class of piecewise-smooth control functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 90–102, November–December 2007.     

Optimization methods for compound poisson risk processes

In this article we introduce the notion of the family of compound Poisson risk processes that depend on a finite-dimensional parameter; we describe examples of such families that arise during formalization of surplus, quota, and quota-surplus reinsurance contracts; the optimization problem is stated for the growth rate of insurance company capital subject to a given constraint on the probability of ruin; numerical methods for solving this problem by Lagrange function saddle-point techniques are reviewed. The research described in this publication was made possible in part by Grant No. UAL000 from the International Science Foundation. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 87–96, March–April, 1998.     

Stability of a vector integer quadratic programming problem with respect to vector criterion and constraints

The paper analyzes five types of stability against perturbations of initial data in criterion functions and constraints of vector integer quadratic optimization problems. Necessary and sufficient conditions are proved for all the types of stability. The relationship among stability with respect to changes of vector criterion coefficients, stability with respect to changes of initial data in constraints, and stability with respect to vector criterion and constraints is established. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 63–72, September–October 2006.     

Self-insurance of investor under repeating catastrophic risks

A decision-making problem of investment into a profitable object in a catastrophic risk area is considered. By a catastrophic risk is meant the probability of severe yet unlikely losses. As a risk hedging mechanism, an insurance fund is considered that is replenished by a part of profit and is used for object renewal. It is shown that methods of insurance mathematics can be used to assess the risk to lose the object. For the plant loss probability as a function of the insurance reserve, integral equations are derived. They can be solved by successive approximations. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 74–83, May–June 2007.     

Optimal control models for interregional migration under social risks

An organizational two-level hierarchic system is used to examine control models for interregional migration under conditions of social risks. The Cauchy problem is used to solve the multidimensional transport equation (Problem A1) and a system of ordinary linear differential equations (Problem A2). For the models proposed, sufficient conditions for the existence of optimal control on the classes of functions designed are established, and stable numeral methods of searching for optimum solutions are developed. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 107–122, May–June 2006.     

Stability criteria in vector combinatorial bottleneck problems in terms of binary relations

The paper considers a vector (multiobjective) path problem with minimax partial criteria. Solving this problem means finding a Pareto set. Binary relations on a set of paths are used to formulate the necessary and sufficient conditions for five types of problem stability against perturbations of the parameters of a vector criterion. Some sufficient stability conditions in terms of Pareto, Smale, and Slater sets are obtained as corollaries. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 103–111, May–June 2008.     

A model of optimal control over a nonlinear multidimensional innovation diffusion process

A model of competitive innovation diffusion is considered. The model is based on the Lotka-Volterra system and an initial-boundary problem for a system of quasilinear parabolic equations. The maximum principle is proved for the problem of diffusion of two competitive innovations, and sufficient conditions of existence of optimum control are obtained for the system. A numerical algorithm is constructed for solving optimum control problems, and numerical results for a model example are presented. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 120–133, July–August 2008.     

Stability of Vector Problems of Integer Optimization: Relationship with the Stability of Sets of Optimal and Nonoptimal Solutions

Several types of stability against perturbations of vector criterion coefficients are analyzed from the same point of view for a vector integer optimization problem with quadratic criterion functions. The concept of stability is defined. Necessary and sufficient conditions are formulated and analyzed for each type of stability. The topological structure of the sets of initial data on which some solution remains optimal is described. The study was supported from the State Fund for Basic Research of Ukraine (Grant F7/275-2001). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 90–100, July–August 2005.     

The leontief-ford dynamic optimization model under conditions of ecological equilibrium

The ecological-economical optimization problem based on the Leontief-Ford model is investigated. Necessary and sufficient conditions of the optimal solution are described for the discrete variant of the model. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 73–79. March–April, 2000.     

On the sufficient equivalence conditions of gaussian measures corresponding to homogeneous fields whose spectral densities have real zeros

The sufficient equivalence criteria of probability measures corresponding to Gaussian homogeneous random fields are given in terms of the spectral densities of the fields considered. These criteria are represented in a form taking into account the possibility of vanishing of the mentioned spectral densities. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 37–48, November–December, 1999.     

Exponential Stability of Uncertain Stochastic Neural Networks with Markovian Switching

This paper is concerned with the exponential stability analysis problem for a class of uncertain stochastic neural networks with Markovian switching. The parameter uncertainties are assumed to be norm bounded. Based on Lyapunov–Krasovskii stability theory and the nonnegative semimartingale convergence theorem, delay-dependent and delay- independent sufficient stability conditions are established. It is also shown that the result in this paper cover some recently published works. Two examples are provided to demonstrate the usefulness of the proposed criteria.     

On uncertainty problems in decision-making

The paper continues studying the uncertainty problem based on [2]. A decision-maker is assumed to have a prevalence relation on the consequences of his actions. The necessary and sufficient existence conditions for uncertainty in a decision-making problem for a nonparametric situation are given. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 116–119, March–April 2008.     

Efficiently solvable cases of quadratic assignment problem with generalized monotonic and incomplete anti-monge matrices

The paper describes new sufficient conditions that provide optimum to a quadratic assignment problem for a given substitution. The conditions separate out sets of matrices in the matrix space. The sets are formed by incomplete Anti-Monge matrices, which permit opposite ordering of rows and columns, and by additive monotone matrices, which are outside the class of Toeplitz matrices. The study was partially supported by INTAS (Project 03-51-5501). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 135–151, January–February 2007.     

Hankel/Toeplitz matrices and the static output feedback stabilization problem

The static output feedback (SOF) stabilization problem for general linear, continuous-time and discrete-time systems is discussed. A few novel necessary and sufficient conditions are proposed, and a modified SOF stabilization problem with performance is studied. For multiple-input single-output (or single-input multiple-output) systems the relation with a class of Hankel matrices, and their inverses, in the continuous-time case and with a class of Toeplitz matrices, in the discrete-time case, is established. These relationships are used to construct conceptual numerical algorithms. Finally, it is shown that, in the continuous-time case, the problem can be recast as a concave–convex programming problem. A few worked out examples illustrate the underlying theory.     

Convergence of a jump procedure in a semi-Markov environment in diffusion-approximation scheme

The sufficient convergence conditions are obtained for a jump stochastic approximation procedure in a semi-Markov environment in a diffusion approximation scheme with balance conditions for a singular perturbation of the regression function. To this end, a singular perturbation problem is solved for the asymptotic representation of the compensating operator of an augmented Markov renewal process. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 124–133, November–December 2007.     

Modeling of a multisector ecological economic system with lags

A model of a diversified ecological economic system with lags is proposed and investigated using sufficient optimality conditions. A turnpike solution of the problem, boundary trajectories, and control switchpoints are found. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 114–122, March–April 2007.