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Parametric analysis of the oscillatory solutions to stochastic differential equations with the Wiener and Poisson components by the Monte Carlo method

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Abstract

Using the Monte Carlo method, we address the influence of the Wiener and Poisson random noises on the behavior of oscillatory solutions to systems of stochastic differential equations (SDEs). For the linear and Van der Pol oscillators, we study the accuracy of estimates of the functionals of numerical solutions to SDEs obtained by the generalized explicit Euler method. For a linear oscillator, we obtain the exact analytical expressions for the mathematical expectation and the variance of the SDE solution. These expressions allow us to investigate the dependence of the accuracy of estimates of the solution moments on the values of SDE parameters, the size of meshsize, and the ensemble of simulated trajectories of the solution. For the Van der Pol oscillator, we study the dependence of the frequency and the damping rate of the oscillations of the mathematical expectation of SDE solution on the values of parameters of the Poisson component. The results of the numerical experiments are presented.

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References

  1. I. I. Gikhman and A. V. Skorokhod, Theory of Stochastic Processes, Vol. 3 (Nauka, Moscow, 1975) [in Russian].

    MATH  Google Scholar 

  2. E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance (Springer, Berlin, 2010).

    Book  MATH  Google Scholar 

  3. R. Mikulevicius and E. Platen, “Time Discrete Taylor Approximations for Ito Processes with Jump Component,” Math. Nachr. 138, 93–104 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  4. D. F. Kuznetsov, Stochastic Differential Equations: Theory and Practice of Numerical Solution (Politekh. Univ., St. Petersburg, 2010) [in Russian].

    Google Scholar 

  5. T. A. Averina, “Methods of Statistical Modeling a Heterogeneous Poisson Ensemble,” Sibirsk. Zh. Vychisl. Mat. 12 (4), 361–374 (2009).

    MATH  Google Scholar 

  6. S. S. Artem’ev, Numerical Solution of Cauchy Problem for a System of Ordinary or Stochastic Differential Equations (Vychisl. Tsentr Sibirsk. Otdel. Ross. Akad. Nauk, Novosibirsk, 1993) [in Russian].

    Google Scholar 

  7. S. S. Artem’ev, A. A. Ivanov, and V. D. Korneev, “Numerical Analysis of Stochastic Oscillators with Supercomputers,” Sibirsk. Zh. Vychisl. Mat. 15 (1), 31–43 (2012).

    MATH  Google Scholar 

  8. T. A. Averina and A. A. Alifirenko, “Stability Analysis of a Linear Oscillator with Multiplicative Noise,” Sibirsk. Zh. Vychisl. Mat. 10 (2), 127–145 (2007).

    MATH  Google Scholar 

  9. T. A. Averina, “Numerical Solution of StochasticDifferential Equations with PoissonComponent,” in Theory and Applications of StatisticalModeling (Vychisl. Tsentr Sibirsk. Otdel. Ross. Akad. Nauk, Novosibirsk, 1989), pp. 81–89.

    Google Scholar 

  10. S.-S. Guo and G.-K. Er, “The Probabilistic Solution of Stochastic Oscillators with Even Nonlinearity under Poisson Excitation,” Central European J. Phys. 10 (3), 702–707 (2012).

    Google Scholar 

  11. V. I. Tikhonov, Statistical Radio Engineering (Radio i Svyaz’, Moscow, 1982) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Institute of Computational Mathematics and Mathematical Geophysics, pr. Akad. Lavrent’eva 6, Novosibirsk, 630090, Russia

    S. S. Artem’ev & M. A. Yakunin

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    S. S. Artem’ev

Authors
  1. S. S. Artem’ev
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  2. M. A. Yakunin
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Correspondence to S. S. Artem’ev.

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Original Russian Text © S.S. Artem’ev, M.A. Yakunin, 2017, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2017, Vol. XX, No. 2, pp. 3–14.

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Artem’ev, S.S., Yakunin, M.A. Parametric analysis of the oscillatory solutions to stochastic differential equations with the Wiener and Poisson components by the Monte Carlo method. J. Appl. Ind. Math. 11, 157–167 (2017). https://doi.org/10.1134/S1990478917020016

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  • DOI: https://doi.org/10.1134/S1990478917020016

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