# Chernetskii V. A.

### Projective method for equation of risk theory in the arithmetic case

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 565-582

We consider a discrete model of operation of an insurance company whose initial capital can take any integer value. In this statement, the problem of nonruin probability is naturally solved by the Wiener-Hopf method. Passing to generating functions and reducing the fundamental equation of risk theory to a Riemann boundary-value problem on the unit circle, we establish that this equation is a special one-sided discrete Wiener-Hopf equation whose symbol has a unique zero, and, furthermore, this zero is simple. On the basis of the constructed solvability theory for this equation, we justify the applicability of the projective method to the approximation of ruin probabilities in the spaces $l^{+}_1$ and $\textbf{c}^{+}_0$. Conditions for the distributions of waiting times and claims under which the method converges are established. The delayed renewal process and stationary renewal process are considered, and approximations for the ruin probabilities in these processes are obtained.

### Algebra of Bergman Operators with Automorphic Coefficients and Parabolic Group of Shifts

Chernetskii V. A., Mozel’ V. A.

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1218-1223

UDC 517.983

We study the algebra of operators with the Bergman kernel extended by isometric weighted shift operators. The coefficients of the algebra are assumed to be automorphic with respect to a cyclic parabolic group of fractional-linear transformations of a unit disk and continuous on the Riemann surface of the group. By using an isometric transformation, we obtain a quasiautomorphic matrix operator on the Riemann surface with properties similar to the properties of the Bergman operator. This enables us to construct the algebra of symbols, devise an efficient criterion for the Fredholm property, and calculate the index of the operators of the algebra considered.

### Unimprovable estimates for solutions of a mixed problem for linear elliptic equations of the second order in a neighborhood of an angular point

Ukr. Mat. Zh. - 1997. - 49, № 11. - pp. 1529–1542

Under the minimal conditions for the smoothness of the coefficients of an equation, unimprovable estimates are obtained for solutions of a mixed problem for linear nondivergent elliptic equations of the second order in a neighborhood of an angular point of the boundary of a domain.

### The Carleman boundary value problem on a Riemannian surface with an edge

Chernetskii V. A., Zverovich E. I.

Ukr. Mat. Zh. - 1970. - 22, № 5. - pp. 591—599