Electronic Communications in Probability

A stochastic scheme of approximation for ordinary differential equations

Raul Fierro and Soledad Torres

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Abstract

In this note we provide a stochastic method for approximating solutions of ordinary differential equations. To this end, a stochastic variant of the Euler scheme is given by means of Markov chains. For an ordinary differential equation, these approximations are shown to satisfy a Large Number Law, and a Central Limit Theorem for the corresponding fluctuations about the solution of the differential equation is proven.

Article information

Source
Electron. Commun. Probab., Volume 13 (2008), paper no. 1, 1-9.

Dates
Accepted: 21 November 2007
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465233428

Digital Object Identifier
doi:10.1214/ECP.v13-1341

Mathematical Reviews number (MathSciNet)
MR2372832

Zentralblatt MATH identifier
1190.60058

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Numerical Scheme Convergence in law Central limit theorem

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Fierro, Raul; Torres, Soledad. A stochastic scheme of approximation for ordinary differential equations. Electron. Commun. Probab. 13 (2008), paper no. 1, 1--9. doi:10.1214/ECP.v13-1341. https://projecteuclid.org/euclid.ecp/1465233428


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References

  • Bykov, A. A. Numerical solution of stiff Cauchy problems for systems of linear ordinary differential equations.(Russian) Vychisl. Metody i Programmirovanie No. 38 (1983), 173–181.
  • Fierro, Raúl; Torres, Soledad. The Euler scheme for Hilbert space valued stochastic differential equations. Statist. Probab. Lett. 51 (2001), no. 3, 207–213.
    Zentralblatt MATH: 0979.60043
    Digital Object Identifier: doi:10.1016/S0167-7152(00)00118-8
  • Henrici, Peter. Discrete variable methods in ordinary differential equations.John Wiley & Sons, Inc., New York-London 1962 xi+407 pp. (24 #B1772)
    Zentralblatt MATH: 0112.34901
  • Kloeden, Peter E.; Platen, Eckhard. Numerical solution of stochastic differential equations.Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. xxxvi+632 pp. ISBN: 3-540-54062-8
    Zentralblatt MATH: 0752.60043
  • Rebolledo, Rolando. Central limit theorems for local martingales. Z. Wahrsch. Verw. Gebiete 51 (1980), no. 3, 269–286.
    Zentralblatt MATH: 0432.60027
    Digital Object Identifier: doi:10.1007/BF00587353
  • San Martín, Jaime; Torres, Soledad. Euler scheme for solutions of a countable system of stochastic differential equations. Statist. Probab. Lett. 54 (2001), no. 3, 251–259.
    Mathematical Reviews (MathSciNet): MR1857939
    Zentralblatt MATH: 0992.60062
    Digital Object Identifier: doi:10.1016/S0167-7152(01)00050-5
  • Srinivasu, P. D. N.; Venkatesulu, M. Quadratically convergent numerical schemes for nonstandard initial value problems. Appl. Math. Comput. 47 (1992), no. 2-3, 145–154.
    Zentralblatt MATH: 0762.65044
    Digital Object Identifier: doi:10.1016/0096-3003(92)90043-Z
  • Wollman, Stephen. Convergence of a numerical approximation to the one-dimensional Vlasov-Poisson system. Transport Theory Statist. Phys. 19 (1990), no. 6, 545–562.
    Zentralblatt MATH: 0728.65121
    Digital Object Identifier: doi:10.1080/00411459008260822

The Institute of Mathematical Statistics and the Bernoulli Society

Electronic Communications in Probability

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