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http://dx.doi.org/10.4134/CKMS.2011.26.4.709

NUMERICAL SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION CORRESPONDING TO CONTINUOUS DISTRIBUTIONS  

Amini, Mohammad (Department of Statistics Ordered and Spatial Data Center of Excellence(OSDCE) Ferdowsi University of Mashhad)
Soheili, Ali Reza (Department of Applied Mathematics The center of excellence in modeling and computations in linear and nonlinear systems(CRMCS) Ferdowsi University of Mashhad)
Allahdadi, Mahdi (Department of Mathematics University of Sistan and Baluchestan)
Publication Information
Communications of the Korean Mathematical Society / v.26, no.4, 2011 , pp. 709-720 More about this Journal
Abstract
We obtain special type of differential equations which their solution are random variable with known continuous density function. Stochastic differential equations (SDE) of continuous distributions are determined by the Fokker-Planck theorem. We approximate solution of differential equation with numerical methods such as: the Euler-Maruyama and ten stages explicit Runge-Kutta method, and analysis error prediction statistically. Numerical results, show the performance of the Rung-Kutta method with respect to the Euler-Maruyama. The exponential two parameters, exponential, normal, uniform, beta, gamma and Parreto distributions are considered in this paper.
Keywords
stochastic differential equation; continuous distribution function; confidence interval; Euler-Maruyama method;
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1 H. P. Langtangen, A general numerical solution method for Fokker-Planck equations with applications to structural relibaility, Probab. Eng. Mech. 6 (1991), no. 1, 33-48.   DOI   ScienceOn
2 X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math. 151 (2003), no. 1, 215-227.   DOI   ScienceOn
3 X. Mao, C. Yuan, and G. Yin, Numerical method for stationary distribution of stochastic differential equations with Markovian switching, J. Comput. Appl. Math. 174 (2005), no. 1, 1-27.   DOI   ScienceOn
4 H. C. Ottinger, Stochastic Processes in Polymeric Fluids, Springer Verlag, Berlin, 1999.
5 M. Di Paola and A. Sofi, Approximate solution of the Fokker-Planck-Kolmogorov equation, Probab. Eng. Mech. 17 (2002), 369-384.   DOI   ScienceOn
6 S. Primak, V. Kontorovich, and V. Lyandres, Stochastic Methods and Their Applications to Communications, John Wiley & Sons Ltd, 2004.
7 H. Risken, The Fokker-Planck Equation, Second Edition. Springer, 1996.
8 Jr. M. S. Torres and J. M. A. Figueiredo, Probability amplitude structure of Fokker-Plank equation, Phys. A 329 (2003), no. 1-2, 68-80.   DOI   ScienceOn
9 A. Tocino and R. Ardanuy, Runge-Kutta methods for numerical solution of stochastic differential equations, J. Comput. Appl. Math. 138 (2002), no. 2, 219-241.   DOI   ScienceOn
10 K. Burrage and P. M. Burrage, Numerical methods for stochastic differential equations with applications, advanced computational modeling center, Unversity of Queensland, Australia, 2002.
11 M. Careltti, Numerical solution of stochastic differential problems in the biosciences, J. Comput. Appl. Math. 185 (2006), no. 2, 422-440.   DOI   ScienceOn
12 J. B. Chen and J. Li, A note on the principle of preservation of probability and probability density evolution equation, Probab. Eng. Mech. Vol. 24 (2009), 51-59.   DOI   ScienceOn
13 S.-N. Chow and H.-M. Zhou, An analysis of phase noise and Fokker-Planck equations, J. Differential Equations 234 (2007), no. 2, 391-411.   DOI   ScienceOn
14 O. Ditlevsen, Invalidity of the spectral Fokker-Planck equation for Cauchy noise driven Langevin equation, Probab. Eng. Mech. 19 (2004), no. 4, 385-392.   DOI   ScienceOn
15 T. D. Frank and A. Daffertshofer, Nonlinear Fokker-Planck equations whose stationary solutions make entropy-like functionals stationary, Phys. A 272 (1999), no. 3-4, 497-508.   DOI   ScienceOn
16 T. D. Frank, A note on the Markov property of stochastic processes described by nonlinear Fokker-Planck equations, Phys. A 320 (2003), no. 1-4, 204-210.   DOI   ScienceOn
17 D. Kim and D. Stanescu, Low-storage Runge-Kutta methods for stochastic differential equations, Appl. Numer. Math. 58 (2008), no. 10, 1479-1502.   DOI   ScienceOn
18 P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlage, Berlin, 1992.
19 Y. Komori, T. Mistsui, and H. Sugiura, Rooted tree analysis of the order continuous of row-type scheme for stochastic differential equation, BIT, 37 (1997), no. 1, 43-66.   DOI
20 L. Ambrosio, G. Savare, and L. Zambotti, Existence and stability for Fokker-Planck equations with log-concave reference measure, Probab. Theory Related Fields 145 (2009), no. 3-4, 517-564.   DOI
21 E. Barkai, Fractional Fokker-Planck equation, solution, and application, Phys. Rev. E 63 (2001), 046118.   DOI
22 E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, J. Comput. Appl. Math. 125 (2000), no. 1-2, 297-307.   DOI   ScienceOn
23 K. Burrage, P. M. Burrage, and T. Tian, Numerical methods for strong solutions of stochastic differential equations: an overview, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460 (2004), no. 2041, 373-402.   DOI   ScienceOn