vestifmИзвестия Национальной академии наук Беларуси. Серия физико-математических наукProceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series1561-24302524-2415The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-2430-2021-57-2-198-205vestifm-586Research ArticleМАТЕМАТИКАMATHEMATICSПриближенные формулы для вычисления математического ожидания функционалов от решения линейного уравнения СкороходаApproximate formulas for the evaluation of the mathematical expectation of functionals from the solution to the linear Skorohod equationЕгоровА. Д.EgorovA. D.

Егоров Александр Дмитриевич – доктор физико-математических наук, главный научный сотрудник

ул. Сурганова, 11, 220072, г. Минск, Республика Беларусь

Alexandr D. Egorov – D r. Sc. ( Physics and Mathematics), Chief Researcher

11, Surganov Str., 220072, Minsk, Republic of Belarus

egorov@im.bas-net.byИнститут математики Национальной академии наук БеларусиInstitute of Mathematics of the National Academy of Sciences of Belarus202115072021572198205Copyright © Егоров А.Д., 20212021Егоров А.Д.Egorov A.D.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestifm.belnauka.by/jour/article/view/586

Данная работа посвящена приближенному вычислению математических ожиданий нелинейных функционалов от решения линейного уравнения Скорохода с ведущим винеровским процессом и случайным начальным условием. Предложен новый подход к построению квадратурных формул, точных для функциональных многочленов третьей степени, который основан на использовании кратных интегралов Стилтьеса. Также построена составная приближенная формула, точная для функциональных многочленов третьего порядка, сходящаяся к точно- му значению ожидания, основанная на комбинации полученной квадратурной формулы и аппроксимации ведущего винеровского процесса. Рассмотрены тестовые примеры, иллюстрирующие применение полученных формул.

This paper is devoted to the construction of approximate formulas for calculating the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod stochastic differential equation with a random initial condition. To calculate the mathematical expectations of nonlinear functionals from random processes, functional analogs of quadrature formulas have been developed, based on the requirement of their accuracy for functional polynomials of a given degree. Most often, formulas are constructed that are exact for polynomials of the third degree [1–9], which are used to obtain an initial approximation and in combination with approximations of the original random process. In the latter case, they are usually also exact for polynomials of a given degree and are called compound formulas. However, in the case of processes specified in the form of compound functions from other random processes the constructed functional quadrature formulas, as a rule, have great computational complexity and cannot be used for computer implementation. This is exactly what happens in the case of functionals from the solutions of stochastic equations. In [1, 2], the approaches to solving this problem were considered for some types of Ito equations in martingales. The solution of the problem is simplified in the cases when the solution of the stochastic equation is found in explicit form: the corresponding approximations were obtained in the cases of the linear equations of Ito, Ito – Levy and Skorohod in [3–11]. In [7, 8, 11], functional quadrature formulas were constructed that are exact for the approximations of the expansions of the solutions in terms of orthonormal functional polynomials and in terms of multiple stochastic integrals. This work is devoted to the approximate calculation of the mathematical expectations of nonlinear functionals from the solution of the linear Skorokhod equation with a leading Wiener process and a random initial condition. A new approach to the construction of quadrature formulas, exact for functional polynomials of the third degree, based on the use of multiple Stieltjes integrals over functions of bounded variation in the sense of Hardy – Krause, is proposed. A composite approximate formula is also constructed, which is exact for second-order functional polynomials, converging to the exact expectation value, based on a combination of the obtained quadrature formula and an approximation of the leading Wiener process. The test examples illustrating the application of the obtained formulas are considered.

стохастические дифференциальные уравненияуравнение Скороходаматематические ожидания функционалов от решенияприближенные формулыstochastic differential equationsSkorochod equationmathematical expectations of functionals from solutionsapproximate formulasReferencesEgorov, A. D. Functional Integrals: Approximate Evaluations and Applications / A. D. Egorov, P. I. Sobolevsky, L. A. Yanovich. – Kluwer Academic Publ., 1993. https://doi.org/10.1007/978-94-011-1761-6Egorov A. D., Sobolevsky P. I., Yanovich L. A. Functional Integrals: Approximate Evaluations and Applications. Kluwer Academic Publishers, 1993. https://doi.org/10.1007/978-94-011-1761-6Егоров, А. Д. Введение в теорию и приложения функционального интегрирования / А. Д. Егоров, Е. П. Жидков, Ю. Ю. Лобанов. – М.: Физматлит, 2006. – 400 с.Egorov A. D., Zhidkov E. P., Lobanov Yu. Yu. Introduction to Theory and Applications of Functional Integration. Moscow, Fizmatlit Publ., 2006. 400 p. (in Russian).Egorov, A. D. 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The authors declare that there are no conflicts of interest present.