Numerical solution of second-order stochastic differential equations with Gaussian random parameters

Subject Areas : Ordinary differential equations

R. Farnoosh ¹ (School of Mathematics, Iran University of Science and Technology, 16844, Tehran, Iran)
H. Rezazadeh ² (School of Mathematics, Iran University of Science and Technology, 16844, Tehran, Iran)
A. Sobhani ³ (School of Mathematics, Iran University of Science and Technology, 16844, Tehran, Iran)
D. Ebrahimibagha ⁴ (Department of Mathematics, Center Branch, Islamic Azad university, Tehran, Iran)

Received: 2013-09-16 Accepted : 2013-12-31 Published : 2014-01-01

Keywords: Linear equations system, damped harmonic oscillators with noise, multiplicative noise, Stochastic Differential Equation, Gaussian random variables,

Abstract :

In this paper, we present the numerical solution of ordinary differential equations(or SDEs), from each order especially second-order with time-varying and Gaussian randomcoefficients. We indicate a complete analysis for second-order equations in special case ofscalar linear second-order equations (damped harmonic oscillators with additive or multiplicative noises). Making stochastic differential equations system from this equation, it couldbe approximated or solved numerically by different numerical methods. In the case of linearstochastic differential equations system by Computing fundamental matrix of this system, itcould be calculated based on the exact solution of this system. Finally, this stochastic equation is solved by numerically method like Euler-Maruyama and Milstein. Also its Asymptoticstability and statistical concepts like expectation and variance of solutions are discussed.

References:

[1] M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Oxford, UK, (1997).

[2] L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, (1974).

[3] K. Burrage, I. Lenane, and G. Lythe, Numerical methods for second-order stochastic differential equations, SIAM J. SCI. Compute., Vol. 29, No. 1, pp. 245264, (2007).

[4] R. Cairoli, J. Walsh, Stochastic integrals in the plane, in Acta Math., 134, pp. 111183., (1975).

[5] Dongbin Xiu, D Daniel M. Tartakovsky, Numerical solution for differential equation in random domain, SIAM J. Sci. Compute. Vol. 28, No. 3, pp. 1167-1185 (2006).

[6] Lawrence C. Evans.:An Introduction to Stochastic Dierential Equations Version 1.2 (2004).

[7] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, 3rd ed., Springer-Verlag, Berlin, (2004).

[8] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review 43, 525-546, (2001).

[9] E. Hairer, S. P. Norsett, and G. Wanner, Solving Ordinary Dierential Equations I: Nonstiff Problems, 2nd ed., Springer-Verlag, Berlin, (1993).

[10] N. V. Krylov, Introduction to the Theory of Diusion Processes, American Math Society, (1995).

[11] J. Lamperti, A simple construction of certain diusion processes, J. Math. Kyoto, 161-170, (1964).

[12] G. N. Milstein and M. V. Tretyakov, Quasi-symplectic methods for Langevin-type equations, IMA J. Numer. Anal., 23, pp. 593626, (2003).

[13] H. McKean, Stochastic Integrals, Academic Press, (1969).

[14] C. A. Marsh and J. M. Yeomans, Dissipative particle dynamics: The equilibrium for nite time steps, Euro-phys. Lett., 37, pp. 511516, (1997).

[15] B. K. Oksendal, Stochastic Dierential Equations: An Introduction with Applications, 4th ed., Springer, (1995).

[16] H. R. Rezazadeha, M. Magasedib, B. Shojaeec.Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation, Journal of Linear and Topological Algebra Vol. 01, No. 02, 79- 89, (2012).

[17] Wuan Luo. Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations. California Institute of Technology Pasadena, California,(2006).