Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

About History Editorial Page Addresses Scope Editorial Staff Submission Review For Authors Publication Ethics Issues Русская версия

Modeling and Analysis of Linear Invariant Stochastic Systems

Author(s):

Tatyana Averina

Institute of Computational Mathematics and Mathematical Geophysics SB RAS;
Novosibirsk State University

ata@osmf.sscc.ru

Elena Karachanskaya

Far Eastern State Transport University,
Pacific National University

elena_chal@mail.ru

Konstantin Rybakov

Moscow Aviation Institute (National Research University)

rkoffice@mail.ru

Abstract:

The main aim of this paper is to test the numerical methods for stochastic differential equations with solutions on a given smooth manifold. Cylindrical surfaces of the second order are selected as manifolds examples for the three-dimensional space (two-dimensional phase space): elliptic, hyperbolic, and parabolic cylinders. We construct classes of stochastic differential equations with solutions on these surfaces and consider linear equations with multiplicative noise. The numerical methods accuracy is estimated by the statistical modeling as mean distance between simulated solutions and the given smooth manifold. These results are compared with a theoretical accuracy of the numerical methods (in the sense of strong convergence).

Keywords

References:

  1. Dubko V. A. Integral invariants for one class of systems of stochastic differential equations. Dopov. Nats. Akad. Nauk Ukr. Mat. Tekh. Nauki, 1984, no. 1, pp. 17-20
  2. Dubko V. A. Voprosy teorii i primeneniya stokhasticheskikh differentsialnykh uravneniy [Problems of Theory and Application of Stochastic Differential Equations]. Vladivostok, Akad. Nauk SSSR, 1989
  3. Karachanskaya E. V. Construction of programmed controls for a dynamic system based on the set of its first integrals. Journal of Mathematical Sciences, 2014, vol. 199, no. 5, pp. 547-555
  4. Karachanskaya E. V. Integralnye invarianty stokhasticheskikh sistem i programmnoe upravlenie s veroyatnostiu 1 [Integral Invariants of Stochastic Systems and Program Control with Probability 1]. Pacific National University, 2015
  5. Averina T. A., Karachanskaya E. V., Rybakov K. A. Statistical modeling of random processes with invariants. Proceedings of the 2017 International Multi-Conference on Engineering, Computer and Information Sciences (SIBIRCON), Novosibirsk, 2017, pp. 34-37
  6. Artemiev S. S., Averina T. A. Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations. VSP, 1997
  7. Averina T. A. Verifikatsiya chislennykh metodov resheniya sistem so sluchainoy strukturoy [Verification of Numerical Methods for Solving Systems with Random Structure]. Novosibirsk, Novosibirsk State University, 2015
  8. Averina T. A. Postroenie algoritmov statisticheskogo modelirovaniya sistem so sluchainoy strukturoy [Construction of Statistical Modeling Algorithms for Systems with Random Structure]. Novosibirsk, Novosibirsk State University, 2015
  9. Averina T. A., Artemiev S. S. A new family of numerical methods for solving stochastic differential equations. Soviet Math. Dokl, 1986, vol. 33, no. 3, pp. 736-738
  10. Averina T. A., Rybakov K. A. Comparison of a statistical simulation method and a spectral method for analysis of stochastic multistructure systems with distributed transitions. Rus. J. Numer. Anal. Math. Modelling, 2007, vol. 22, no. 5, pp. 431-447
  11. Burrage K., Tian T. Predictor-corrector methods of Runge-Kutta type for stochastic differential equations. SIAM J. Numer. Anal, 2002, vol. 40, no. 4, pp. 1516-1537
  12. Burrage K., Burrage P. M., Tian T. Numerical methods for strong solutions of stochastic differential equations: an overview. Proc. R. Soc. Lond. A, 2004, vol. 460, no. 2041, pp. 373-402
  13. Kloeden P. E., Pearson R. A. The numerical solution of stochastic differential equations. J. Aust. Math. Soc. B, 1977, vol. 20, pp. 8-12
  14. Kloeden P. E., Platen E. Numerical Solution of Stochastic Differential Equations. Springer, 1995
  15. Kuznetsov D. F. Multiple Ito and Stratonovich stochastic integrals: Fourier-Legendre and trigonometric expansions, approximations, formulas. Differential Equations and Control Processes, 2017, no. 1
  16. Maruyama G. Continuous Markov processes and stochastic equations. Rend. Circolo Math. Palermo, 1955, vol. 2, no. 4, pp. 48-90
  17. Milshtein G. N. Approximate integration of stochastic differential equations. Theory Probab. Appl., 1974, vol. 19, no. 3, pp. 557-562
  18. Milstein G. N., Tretyakov M. V. Stochastic Numerics for Mathematical Physics. Springer, 2004
  19. Nikitin N. N., Razevig V. D. Digital simulation of stochastic differential equations and error estimates. USSR Comput. Math. Math. Phys., 1978, vol. 18, no. 1, pp. 102-113
  20. Panteleev A. V., Rudenko E. A., Bortakovskiy A. S. Nelineynye sistemy upravleniya: opisanie, analiz i sintez [Nonlinear Control Systems: Description, Analysis, and Synthesis]. Moscow, University Book, 2008
  21. Saul’ev V. K. Chislennoe reshenie uravneniy sluchainykh protcessov [Numerical Solution of Random Processes Equations]. Moscow, MAI, 1989
  22. Gihman I. I., Skorohod A. V. Stochastic Differential Equations. Springer, 1972

Full text (pdf)