Iнварiантнi поверхнi для певних класiв систем другого порядку стохастичних диференцiальних рiвнянь зi стрибками

In this paper, we consider the concept of invariant sets of inhomogeneous stochastic differential equations with jumps. For certain classes of systems of the second order of inhomogeneous stochastic differential equations with jumps the necessary and sufficient conditions for the invariance of the corresponding surfaces are established. The obtained results provide opportunities to find the invariant surfaces and conditions of their invariance for the specified classes of stochastic differential equations. Pages of the article in the issue: 22 - 27 Language of the article: UkrainianУ статтi розглядається поняття iнварiантних множин неоднорiдних стохастичних диференцiальних рiвнянь зi стрибками. Для певних класiв систем другого порядку неоднорiдних стохастичних диференцiальних рiвнянь зi стрибками знайдено необхiднi i достатнi умови iнварiантностi вiдповiдних поверхонь. Отриманi результати дозволяють знаходити iнварiантнi поверхнi та умови їх iнварiантностi для вказаних класiв стохастичних диференцiальних рiвнянь. Сторінки у випуску: 11 - 21 Мова статті: українськ

Наближення дробових iнтегралiв гельдерових функцiй

The paper is devoted to the rate of convergence of integral sums of two different types to fractional integrals. The first theorem proves the H¨older property of fractional integrals of functions from various integral spaces. Then we estimate the rate of convergence of the integral sums of two types corresponding to the H¨older functions, to the respective fractional integrals. We illustrate the obtained results by several figures. Pages of the article in the issue: 18 - 25 Language of the article: EnglishСтаттю присвячено дослiдженню швидкостi збiжностi iнтегральних сум двох типiв до дробового iнтегралу. В першiй теоремi доведено гельдерiвську властивiсть дробових iнтегралiв вiд функцiй з рiзних iнтегральних просторiв. Потiм ми оцiнюємо швидкiсть збiжностi iнтегральних сум, побудованих за гельдерiвськими функцiями, до вiдповiдних дробових iнтегралiв. Отриманi результати проiлюстровано декiлькома чисельними прикладами

Мiжнародна наукова конференцiя “Сучасна стохастика: теорiя та застосування. V” (MSTA-V). 1-4 червня 2021

Pages of the article in the issue: 9 Language of the article: UkrainianPages of the article in the issue: 9 Language of the article: Ukrainia

Професор Г.Л. Кулініч (09.12.1938 – 10.02.2022) – видатний вчений і педагог

Pages of the article in the issue: 11 - 21 Language of the article: UkrainianСторінки у випуску: 11 - 21 Мова статті: українськ

Geometry and field theory in multi-fractional spacetime

We construct a theory of fields living on continuous geometries with fractional Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski spacetime. After reviewing the properties of fractional spaces with fixed dimension, presented in a companion paper, we generalize to a multi-fractional scenario inspired by multi-fractal geometry, where the dimension changes with the scale. This is related to the renormalization group properties of fractional field theories, illustrated by the example of a scalar field. Depending on the symmetries of the Lagrangian, one can define two models. In one of them, the effective dimension flows from 2 in the ultraviolet (UV) and geometry constrains the infrared limit to be four-dimensional. At the UV critical value, the model is rendered power-counting renormalizable. However, this is not the most fundamental regime. Compelling arguments of fractal geometry require an extension of the fractional action measure to complex order. In doing so, we obtain a hierarchy of scales characterizing different geometric regimes. At very small scales, discrete symmetries emerge and the notion of a continuous spacetime begins to blur, until one reaches a fundamental scale and an ultra-microscopic fractal structure. This fine hierarchy of geometries has implications for non-commutative theories and discrete quantum gravity. In the latter case, the present model can be viewed as a top-down realization of a quantum-discrete to classical-continuum transition.Comment: 1+82 pages, 1 figure, 2 tables. v2-3: discussions clarified and improved (especially section 4.5), typos corrected, references added; v4: further typos correcte

Mixed Power Variations with Statistical Applications

PLENARY LECTURE

Professor Yu.V. Kozachenko (01.12.1940 - 05.05.2020) - prominent scientist and teacher

Pages of the article in the issue: 9 - 29Language of the article: Ukrainia

Pricing by hedging and no-arbitrage beyond semimartingales

Arbitrage, Pricing, Quadratic variation, Robust hedging, G10, G13,