Numerical integration of Heath-Jarrow-Morton model of interest rates

We propose and analyze numerical methods for the Heath-Jarrow-Morton (HJM) model. To construct the methods, we first discretize the infinite dimensional HJM equation in maturity time variable using quadrature rules for approximating the arbitrage-free drift. This results in a finite dimensional system of stochastic differential equations (SDEs) which we approximate in the weak and mean-square sense using the general theory of numerical integration of SDEs. The proposed numerical algorithms are computationally highly efficient due to the use of high-order quadrature rules which allow us to take relatively large discretization steps in the maturity time without affecting overall accuracy of the algorithms. Convergence theorems for the methods are proved. Results of some numerical experiments with European-type interest rate derivatives are presented.Comment: 48 page

Intellectual adaptive system of computer education

Conceptions of intellectual adaptive system of computer education are described

Current Financial Diagnostics of Enterprises

In this paper we are discussing the concepts and components of the Current Financial Diagnostics of Enterprises that is being developed at the Belarusian State University. Such system can be successfully employed either for training experts in financial analytics and financial management or for financial managers and financial directors at an enterprise for the effective financial decision making

Обобщение теорем Ролля и Дарбу для функций двух переменных

As well known, the classical Rolle and Darboux theorems for a function of one variable establish the existence of a critical point in the behavior of a function at the ends of a closed interval. The question arises of the possibility of extension of the Rolle and Darboux theorems to functions of two variables. More precisely is the existence of a critical point in Ω̅ determined by the behavior of the function f on the boundary of the ∂Ω domain Ω. As shown by A. I. Perov, such extension can be obtained with the help of the concept of rotation. In this article, we establish deeper relations between the Rolle and Darboux theorems and the rotation of a vector field on the boundary ∂Ω. We also present some new formulas for calculating the rotation of a vector field on the boundary, on the basis of which statements about the existence of critical points are formulated.Как известно, классические теоремы Ролля и Дарбу для функции одной переменной устанавливают существование критической точки по поведению функции на концах отрезка. Возникает вопрос о возможности переноса теорем Ролля и Дарбу для функций двух переменных. Более точно, определяется ли существование критической точки в Ω̅ по поведению функции f на границе ∂Ω области Ω. Как было показано А. И. Перовым, такие обобщения можно получить с помощью понятия вращения. В настоящей работе устанавливаются более глубокие связи между теоремами Ролля, Дарбу и вращением векторного поля на границе  ∂Ω. Также приводятся некоторые новые формулы для вычисления вращения векторного поля на границе ∂Ω, на основе которых сформулированы утверждения о существовании критических точек

Об индексе Пуанкаре плоских полиномиальных полей третьей и четвертой степени

The conditions of isolation of a zero singular point of plane polynomial fields of third and fourth degree are considered in terms of the coefficients of the components of these fields. The isolation conditions depend on the greatest common divisor of the components of polynomial fields: in some cases only on its degree, and in some cases, additionally, on the presence of nonzero real zeros. The reasoning, which allows one to write out the isolation conditions, is based on the concept of the resultant and subresultants of components of plane polynomial fields. If the zero singular point is isolated, its index is calculated through the values of subresultants and coefficients of components.Выписываются условия изолированности нулевой особой точки плоских полиномиальных полей третьей и четвертой степени через коэффициенты компонент этих полей. Как оказалось, данные условия существенно зависят от наибольшего общего делителя компонент плоских полиномиальных полей: в некоторых случаях только от его степени, а в некоторых – дополнительно от наличия у него ненулевых вещественных нулей. Соот вет ствующие рассуждения строятся на понятии результанта и субрезультантов компонент поля. В случае изолированности особой точки для ее индекса предлагаются достаточно простые формулы через субрезультанты и коэффициенты компонент

Improving Josephson junction reproducibility for superconducting quantum circuits: junction area fluctuation

Josephson superconducting qubits and parametric amplifiers are prominent examples of superconducting quantum circuits that have shown rapid progress in recent years. With the growing complexity of such devices, the requirements for reproducibility of their electrical properties across a chip have become stricter. Thus, the critical current $I_c$ variation of the Josephson junction, as the most important electrical parameter, needs to be minimized. Critical current, in turn, is related to normal-state resistance the Ambegaokar-Baratoff formula, which can be measured at room temperature. Here, we focus on the dominant source of Josephson junction critical current non-uniformity junction area variation. We optimized Josephson junctions fabrication process and demonstrate resistance variation of $9.8-4.4\%$ and $4.8-2.3\%$ across $22{\times}22$ $mm^2$ and $5{\times}10$ $mm^2$ chip areas, respectively. For a wide range of junction areas from $0.008$ ${\mu}m^2$ to $0.12$ ${\mu}m^2$ we ensure a small linewidth standard deviation of $4$ $nm$ measured over 4500 junctions with linear dimensions from $80$ to $680$ $nm$. The developed process was tested on superconducting highly coherent transmon qubits $(T_1 > 100\:{\mu}s)$ and a nonlinear asymmetric inductive element parametric amplifier