# Volume 59, № 11, 2007

### Estimation of the ruin probability of an insurance company operating on a BS-market

Androshchuk M. O., Mishura Yu. S.

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1443–1453

We obtain an estimate for the ruin probability of an insurance company that invests a part of its capital in stocks and puts the rest of the capital in a bank account. An insurance premium is established depending on the capital of the insurance company.

### Nonexplosion and solvability of nonlinear diffusion equations on noncompact manifolds

Antoniouk A. Val., Antoniouk A. Vict.

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1454–1472

We find sufficient conditions on coefficients of diffusion equation on noncompact manifold, that guarantee non-explosion of solutions in a finite time.
This property leads to the existence and uniqueness of solutions for corresponding stochastic differential equation with globally non-Lipschitz coefficients.

Proposed approach is based on the estimates on diffusion generator, that weakly acts on the metric function of manifold.
Such estimates enable us to single out a manifold analogue of monotonicity condition on the joint behaviour of the curvature of manifold and coefficients of equation.

### Behavior of risk processes with random premiums after ruin and a multivariate ruin function

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1473–1484

We establish relations for the distribution of functionals associated with the behavior of a risk process with random premiums after ruin and for a multivariate ruin function.

### Two-boundary problems for a random walk

Kadankov V. F., Kadankova T. V., Yezhov I. I.

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1485–1509

We solve main two-boundary problems for a random walk. The generating function of the joint distribution of the first exit time of a random walk from an interval and the value of the overshoot of the random walk over the boundary at exit time is determined. We also determine the generating function of the joint distribution of the first entrance time of a random walk to an interval and the value of the random walk at this time. The distributions of the supremum, infimum, and value of a random walk and the number of upward and downward crossings of an interval by a random walk are determined on a geometrically distributed time interval. We give examples of application of obtained results to a random walk with one-sided exponentially distributed jumps.

### Finding cocycles in the bicrossed product construction for Lie groups

Chapovsky Yu., Kalyuzhnyi A. A., Podkolzin G. B.

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1510–1522

We find an explicit formula for finding pairs of cocycles for the construction of examples of locally compact quantum groups by using the bicrossed product of Lie groups.

### Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables

Lavrenyuk S. P., Pukach P. Ya.

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1523–1531

We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain unbounded with respect to space variables. We consider the case of an arbitrary number of space variables. We obtain conditions for the existence and uniqueness of the solution of this problem independent of the behavior of solution as $|x| \rightarrow +\infty$. The indicated classes of the existence and uniqueness are defined as spaces of local integrable functions. The dimension of the domain in no way limits the order of nonlinearity.

### Group classification of quasilinear elliptic-type equations. I. Invariance with respect to Lie algebras with nontrivial Levi decomposition

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1532–1545

The problem of group classification of quasilinear elliptic-type equations in the two-dimensional space is considered. The list of all equations of this sort is obtained, which admit the semisimple Lie algebras of symmetry operators and the Lie algebras of symmetry operators with nontrivial Levi decomposition.

### Investigation of one convective Stefan problem by the Ritz method

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1546–1556

A plane stationary convective Stefan problem is analyzed in the case where the convection is caused by the presence of a prescribed rotation of intensity μ. A method of studying this problem is proposed which consists in a series expansion of the solution in terms of powers of a small parameter μ. The null expansion term is defined by the Rietz method. The formula describing the dependence of free boundary equation on μ is obtained.

### Coherentization of the energy of heat fluctuations by a two-channel bilinear control system

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1557–1573

We propose and investigate a mathematical model of an open bilinear control system for the conversion of heat energy in a coherent form. We show that the use of a combinational parametric resonance formed by the control system in a one-temperature ensemble of weakly dissipative elastic-gyroscopic subsystems enables one to obtain a positive energy output without using any cooling device apart from the control system.

### Properties of parabolic Kählerian spaces admitting an almost geodesic mapping of the type π_{2 } with degenerate affinor structure

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1574–1579

We study an almost geodesic mapping of Riemann spaces with parabolic affinor structure. Some properties of parabolic Kählerian spaces admitting an almost geodesic mapping are established.

### Inverse scattering problem for a wave equation with absorption

Ukr. Mat. Zh. - 2007. - 59, № 11. - pp. 1580–1584

We prove a uniqueness theorem for the inverse scattering problem for a wave equation with absorption and develop an algorithm for the solution of this problem on the basis of a given scattering operator.