Volume 10, Issue 2

In this paper theorems which give conditions for a uniform space to have an ARU -resolution are proved. In particular, a finitistic uniform space admits an ARU -resolution if and only if it has trivial uniform shape or it is an absolute uniform shape retract.

Functions of dimension modulo a (rather wide) class of spaces are considered and the conditions are found, under which the dimension of the product of spaces modulo these classes is equal to zero. Based on these results, the sufficient conditions are established, under which spaces of free topological semigroups (in the sense of Marxen) and spaces of free topological groups (in the sense of Markov and Graev) are zero-dimensional modulo classes of compact spaces.

In this paper we study the approximation by rectangular partial sums of a double Fourier series with respect to the Walsh–Kaczmarz system in the spaces 𝐶 and 𝐿. From our results we obtain different criteria of the uniform convergence and 𝐿-convergence of a double Fourier–Kaczmarz series.

We consider the mean-variance hedging problem in the discrete time setting. Using the dynamic programming approach we obtain recurrent equations for an optimal strategy. Additionally, some technical restrictions of the previous works are removed.

A method of extending σ -finite quasi-invariant measures given on an uncountable group, by using a certain family of its subgroups, is investigated.

Basic static boundary value problems of elasticity are considered for a semi-infinite curvilinear prism Ω = { ρ 0 < ρ < ρ 1 , α 0 < α < α 1 , 0 < 𝑧 < ∞} in generalized cylindrical coordinates ρ , α , 𝑧 with Lamé coefficients ℎ ρ = ℎ α = ℎ( ρ , α ), ℎ 𝑧 = 1. It is proved that the solution of some boundary value problems of elasticity can be reduced to the sum of solutions of other boundary value problems of elasticity. Besides its cognitive significance, this fact also enables one to solve some non-classical elasticity problems.

A multidimensional parametric filtered statistical model is considered, the notions of the regularity and ergodicity of the model, as well as the notion of a general 𝑀-estimator are studied. A skew projection technique is proposed, by which we construct an 𝑀-estimator of the parameter of interest in the presence of nuisance parameters, which is asymptotically normal with the zero mean and the same covariance matrix as the corresponding block of the asymptotic covariance matrix of the full parameter's 𝑀-estimator.

We give a unified characterization of 𝑞-optimal martingale measures for 𝑞 ∈ [0, ∞) in an incomplete market model, where the dynamics of asset prices are described by a continuous semimartingale. According to this characterization the variance-optimal, the minimal entropy and the minimal martingale measures appear as the special cases 𝑞 = 2, 𝑞 = 1 and 𝑞 = 0 respectively. Under assumption that the Reverse Hölder condition is satisfied, the continuity (in 𝐿 1 and in entropy) of densities of 𝑞-optimal martingale measures with respect to 𝑞 is proved.

We consider vector-valued random fields on a general index set in and show the existence of continuous modifications if some Kolmogorov type condition is satisfied. Furthermore, we prove an extension result for Hölder continuous vector valued mappings.

We present a random probability distribution which approximates, in the sense of weak convergence, the Dirichlet process and supports a Bayesian resampling plan called a proper Bayesian bootstrap.

The problem of pricing of time-dependent barrier options is considered in the case when interest rate and volatility are given functions in Black–Scholes framework. The calculation of the fair price reduces to the calculation of non-linear boundary crossing probabilities for a standard Brownian motion. The proposed method is based on a piecewise-linear approximation for the boundary and repeated integration. The numerical example provided draws attention to the performance of suggested method in comparison to some alternatives.

We consider the problem of transition from a weakly separated family of probability measures to a strictly separated family.

Using the Riccati transformation techniques, we establish some new oscillation criteria for the second-order nonlinear difference equation Some comparison between our theorems and the previously known results in special cases are indicated. Some examples are given to illustrate the relevance of our results.

We discuss an application of an inequality for the modulus of the characteristic function of a system of monomials in random variables to the convergence of the density of the corresponding system of the sample mixed moments. Also, we consider the behavior of constants in the inequality for the characteristic function of a trigonometric analogue of the above-mentioned system when the random variables are independent and uniformly distributed. Both inequalities were derived earlier by the author from a multidimensional analogue of Vinogradov's inequality for a trigonometric integral. As a byproduct the lower bound for the spectrum of is obtained, where 𝐴 𝑘 is the matrix of coefficients of the first 𝑘 + 1 Chebyshev polynomials of first kind.

Some well-known properties of the trigonometric system as well as of the Haar and Welsh systems are generalized to general orthonormal systems.

We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.