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"10.29235/1561-2430-2019-55-2-158-168vestifm-382Research ArticleМАТЕМАТИКАMATHEMATICSО составных формулах для математического ожидания функционалов отрешения уравнения Ито в гильбертовом пространствеOn composite formulas for mathematical expectation of functionals of solution of the Ito equation in Hilbert spaceЕгоровА. Д.EgorovA. D.

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

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

Alexandr D. Egorov – Dr. 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 Belarus201928062019552158168Copyright © Егоров А.Д., 20192019Егоров 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/382

Данная работа посвящена построению составных приближенных формул для вычисления математического ожидания нелинейных функционалов от решения линейного уравнения Ито в гильбертовом пространстве с аддитивным шумом. В качестве ведущего процесса рассматривается винеровский процесс, принимающий значения в гильбертовом пространстве. Формулы представляют собой сумму аппроксимаций нелинейных функционалов, полученных разложением ведущего случайного процесса в ряд независимых гауссовских случайных величин, и корректирующих аппроксимирующих функциональных квадратурных формул, обеспечивающих точность составных формул для полиномов третьего порядка. В качестве тестового примера рассмотрено применение полученных формул к случаю одномерного по пространственной переменной волнового уравнения с ведущим винеровским процессом, индексированным пространственной и временной переменными.

This article is devoted to constructing composite approximate formulas for calculation of mathematical expectation of nonlinear functionals of solution of the linear Ito equation in Hilbert space with additive noise. As the leading process, the Wiener process taking values in Hilbert space is examined. The formulas are a sum of the approximations of the nonlinear functionals obtained by expanding the leading random process into a series of independent Gaussian random variables and correcting approximating functional quadrature formulas that ensure an approximate accuracy of compound formulas for third-order polynomials. As a test example, the application of the obtained formulas to the case of a one-dimensional wave equation with a leading Wiener process indexed by spatial and temporal variables is considered. This article continues the research begun in [1].

The problem is motivated by the necessity to calculate the nonlinear functionals of solution of stochastic partial differential equations. Approximate evaluation of mathematical expectation of stochastic equations with a leading random process indexed only by the time variable is considered in [2–11]. Stochastic partial equations in various interpretations are considered [12–16]. The present article uses the approach given in [12].

стохастические уравнения в гильбертовом пространствематематические ожидания функционалов от решенийсоставные приближенные формулыстохастическое волновое уравнениеstochastic differential equations in Hilbert spacemathematical expectation of functionals of solutioncomposite approximate formulasReferencesЕгоров, А. Д. Приближенные формулы для вычисления математического ожидания функционалов от решения уравнения Ито в гильбертовом пространстве / А. Д. Егоров // Докл. Нац. акад. наук Беларуси. – 2016. – т. 60, № 6. – С. 7–13.Egorov A. D. Approximate formulas for calculating the mathematical expectation of functionals of solution of the Ito equations in a Hilbert space. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2016, vol. 60, no. 6, pp. 7–13 (in Russian).Egorov, A. D. Functional integrals: Approximate evaluations and applications / A. D. Egorov, P. I. Sobolevsky, L. A. Yanovich. – Kluwer Academic Publishers, 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/10.1007/978-94-011-1761-6Kloeden, P. E. Numerical Solution of Stochastic Differential Equations / P. E. Kloeden, E. Platen. – Berlin: Springer Science & Business Media, 2013. – 636 p.Kloeden P. E., Platen E. Numerical Solution of Stochastic Differential Equations. Berlin, Springer Science & Business Media, 2013. 636 p.Егоров, 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. Approximations of functional integrals with respect to measure generated by solutions of stochastic differential equations / A. D. Egorov, A. V. Zherelo // Monte Carlo Methods and Applications. – 2004. – Vol. 10, № 3/4. – P. 257–264. https://doi.org/10.1515/mcma.2004.10.3-4.257Egorov A. D., Zherelo A. V. Approximations of functional integrals with respect to measure generated by solutions of stochastic differential equations. Monte Carlo Methods and Applications, 2004, vol. 10, no. 3–4, pp. 257–264. https://doi.org/10.1515/mcma.2004.10.3-4.257Egorov, A. D. Approximations for expectation of functionals of solutions to stochastic differential equations / A. D. Egorov // Monte Carlo Methods and Applications. – 2007. – Vol. 13, № 4. – P. 275–185. https://doi.org/10.1515/mcma.2004.10.3-4.257Egorov A. D. Approximations for expectation of functionals of solutions to stochastic differential equations. Monte Carlo Methods and Applications, 2007, vol. 13, no. 4, pp. 275–185. https://doi.org/10.1515/mcma.2004.10.3-4.257Egorov, A. D. Approximate formulas for expectation of functionals of solutions to stochastic differential equations / A. D. Egorov, K. K. Sabelfeld // Monte Carlo Methods and Applications. – 2010. – Vol. 16, № 2. – P. 95–127. https://doi.org/10.1515/mcma.2010.003Egorov A. D., Sabelfeld K. K. Approximate formulas for expectation of functionals of solutions to stochastic differential equations. Monte Carlo Methods and Applications,2010, vol. 16, no. 2, pp. 95–127. https://doi.org/10.1515/mcma.2010.003Milstein, G. N. Evaluation of conditional Wiener integrals by numerical integration of stochastic differential equations / G. N. Milstein, M. V. Tretyakov // J. Comput. Phys. – 2004. – Vol. 197, № 1. – P. 275–298. https://doi.org/10.1016/j.jcp.2003.12.001Milstein G. N., Tretyakov M. V. Evaluation of conditional Wiener integrals by numerical integration of stochastic differential equations, Journal of Computational Physics, 2004, vol. 197, no. 1, pp. 275–298. https://doi.org/10.1016/j.jcp.2003.12.001Dumas, W. M. Computing conditional Wiener integrals of functional of a general form / W. M. Dumas, M. V. Tretyakov // IMA J. Numerical Analysis. – 2011. – Vol. 31, № 3. – P. 1217–1251. https://doi.org/10.1093/imanum/drq008Dumas W. M., Tretyakov M. V. Computing conditional Wiener integrals of functional of a general form. IMA Journal of Numerical Analysis, 2011, vol. 31, no. 3, pp. 1217–1251. https://doi.org/10.1093/imanum/drq008Применение функциональных интегралов к стохастическим уравнениям / Э. А. Айрян [и др.] // Мат. моделирование. – 2016. – т. 28, № 11. – С. 113–125.Airjan E. A., Egorov A. D., Kulyabov D. S., Malyutin V. B., Sevastyanov L. A. Application of functional integrals to stochastic equations. Matematicheskoe modelirovanie Mathematical Models and Computer Simulations,2017, vol. 9, no. 3, pp. 339–348. https://doi.org/10.1134/s2070048217030024Egorov, A. A method for the calculation of characteristics for the solution to stochastic differential equations / A. Egorov, V. Malyutin // Monte Carlo Methods and Applications. – 2017. – Vol. 23, № 3. – P. 149–157. https://doi.org/10.1515/mcma-2017-0110Egorov A., Malyutin V. A method for the calculation of characteristics for the solution to stochastic differential equations. Monte Carlo Methods and Applications,2017, vol. 23, no. 3, pp. 149–157. https://doi.org/10.1515/mcma-2017-0110Da Prato, G. Stochastic Equations in Infinite Dimensions / G. Da Prato, J. Zabczyk. – Cambridge University Press, 1992. – 454 p. https://doi.org/10.1017/cbo9780511666223Da Prato G., Zabczyk J. Stochastic Equations in Infinite Dimensions. Cambridge University Press, 1992. 454 p. https://doi.org/10.1017/cbo9780511666223Gavarecki, G. V. Stochastic Differential Equations / G. V. Gavarecki, V. Mandrekar // Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations. – Berlin: Springer, 2011. – P. 73–149. https://doi.org/10.1007/978-3-642-16194-0_3Gavarecki G. V., Mandrekar V. Stochastic Differential Equations. Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations. Berlin, Springer, 2011, pp. 73–149. https://doi.org/10.1007/978-3-642-16194-0_3A Minicourse on Stochastic Partial Differential Equations / R. S. Dalang [et al.]. – Springer, 2006. – 222 p.Dalang R. S., Khoshnevisan D., Mueller C., Nualart D., Xiao Y. A Minicourse on Stochastic Partial Differential Equations. Springer, 2006. 222 p.Hairer, M. An Introduction to Stochastic PDEs / M. Hairer. – The University of Warwick/Courant Institute, 2009. – 78 p.Hairer M. An Introduction to Stochastic PDEs. The University of Warwick/Courant Institute, 2009. 78 p.Jentzen, A. Taylor Approximations for Stochastic Partial Differential Equations / A. Jentzen, P. E. Kloeden. – Philadelphia: SIAM Press, 2011. – 235 p. https://doi.org/10.1137/1.9781611972016Jentzen A., Kloeden P. E. Taylor Approximations for Stochastic Partial Differential Equations. Philadelphia, SIAM Press, 2011. 235 p. https://doi.org/10.1137/1.9781611972016Лиходед, Н. А. Уточнение монте-карловской оценки континуальных интегралов / Н. А. Лиходед // Вес. Акад. навук БССР. Сер. фiз.-мат. навук. – 1990. – № 2.– С. 8–13.Likhoded N. A. Refinement of the Monte Carlo estimate of continual integrals. Vescì Akademìì navuk BSSR. Seryâ fìzìka-matèmatyčnyh navuk [Proceedings of the Academy of Sciences of BSSR. Physics and Mathematics Series], 1990, no. 2, pp. 8–13 (in Russian).

The authors declare that there are no conflicts of interest present.