Abstract
In this paper, an adaptive weak scheme for stochastic delay differential equations (SDDEs) based on the weak continuous Euler-Maruyama method which is a special member of the family of continuous weak Runge-Kutta schemes is introduced. The framework of the analysis of the global error is to embed the SDDE into a series of interrelated SDEs each defined on a separate interval in order to consider the error of SDE method and that of the interpolation. We perform the error estimation in a priori form based on the rooted tree theory of Rößler and then analyze the global error of the scheme by obtaining a computable expression of the principal terms of that which is useful for controlling it which contains both the numerical and statistical errors. Adopting the idea presented in Szepessy et al. (Commun. Pure Appl. Math. 54:1169–1214, 2001), we determine the optimal discretization points using the deterministic time-step mechanism and also the necessary number of realizations based on the standard deviation of the approximate solution. We show that this technique leads to increased accuracy of the expected value of the required functionals. By presenting some numerical experiments, the effectiveness of utilizing the adaptive idea with holding the tolerance proportionality property is illustrated.
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Bahar, A., Mao, X.: Stochastic delay population dynamics. Int. J. Pure Appl. Math. 11(4), 377–399 (2004)
Baker, C.T.H., Buckwar, E.: Numerical analysis of explicit one-step methods for stochastic delay diffrential equations. LMS J. Comput. Math. 3, 315–335 (2000)
Bellen, A., Zennaro, M.: Numerical Methods for Delay Diffrential Equations. Oxford University Press (2003)
Beuter, A., Vasilakos, K.: Effects of noise on a delayed visual feedback system. J. Theor. Biol. 165, 389–407 (1993)
Buckwar, E., Mohammed, S.E.A., Kuske, R., Shardlow, T.: Weak convergence of the Euler scheme for stochastic differential delay equations. LMS J. Comput. Math. 11, 60–99 (2008)
Buckwar, E., Shardlow, T.: Weak approximation of stochastic differential delay equations. IMA J. Numer. Anal. 25, 57–86 (2005)
Corwin, S.P., Thompson, S.: Error estimation and step size control for delay differential equation solvers based on continuously embedded Runge-Kutta-Sarafyan methods. Comput. Math. Appl. 31(6), 1–11 (1996)
Debrabant, K., Rössler, A.: Classification of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations. Math. Comput. Simul. 77, 408–420 (2008)
Debrabant, K., Rössler, A.: Continuous weak approximation for stochastic differential equations. J. Comput. Appl. Math. 214, 259–273 (2008)
Durrett, R.: Probability: Theory and Example. Duxbury Press, Belmont, Calif (1964)
Filippi, S., Buchacker, U.: Stepsize control for delay differential equations using a pair of formulae. J. Comput. Appl. Math. 26, 339–343 (1989)
Foroush Bastani, A., Hosseini, S.M.: A new adaptive Runge-Kutta method for stochastic differential equations. J. Comput. Appl. Math. 206, 631–644 (2007)
Gaines, J.G., Lyons, T.J.: Variable step size control in the numerical solution of stochastic differential equations. SIAM J. Appl. Math. 57(5), 1455–1484 (1997)
Hu, Y., Mohammed, S.E.A., Yan, F.: Discrete-time approximation of stochastic delay equations: The Milstein scheme. Ann. Probab. 32, 265–314 (2004)
Kazmerchuk, Y., Swishchuk, A., Wu, J.: The pricing of options for securities markets with delayed response. Math. Comput. Simul. 75(3-4), 69–79 (2007)
Kemajou, E.: A Stochastic Delay Model for Pricing Corporate Liabilities, Ph.D Thesis. Southern Illinois University, Carbondale (2012)
Kpper, D., Lehn, J., Rössler, A.: A step size control algorithm for the weak approximation of stochastic differential equations. Numer. Algorithm. 44, 335–346 (2007)
Küchler, U., Platen, E.: Strong discrete time approximation of stochastic diffrential equations with time delay. Math. Comput. Simul. 54, 189–205 (2000)
Küchler, U., Platen, E.: Weak discrete time approximation of stochastic diffrential equations with time delay. Math. Comput. Simul. 59, 497–507 (2002)
Kushner, H.: Numerical Methods for Controlled Stochastic Delay Systems. Birkhauser (2008)
Lamba, H., Mattingly, J.C., Stuart, A.M.: An adaptive Euler-Maruyama scheme for SDEs: convergence and stability. IMA J. Numer. Anal. 27, 479–506 (2007)
Liang, C., Lord, G.: Stochastic Methods in Neuroscience. Oxford University Press, New York (2010)
Mao, X: Numerical solutions of stochastic functional diffrential equations. LMS J. Comput. Math. 6, 141–161 (2003)
Mao, X. : Stochastic Differential Equations and Applications. Horwood Publishing, Chichester, UK (1997)
Mauthner, S.: Stepsize control in the numerical solution of stochastic differential equations. J. Comput. Appl. Math. 100, 93–109 (1998)
Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Kluwer Academic Publishers, Dordrecht (1995)
Mohammed, S.E.A., Scheutzow, M.K.R.: Lyapunov exponents of linear stochastic functional differential equations, Part II: Examples and case studies. Ann. Probab. 25(3), 1210–1240 (1997)
Moon, K.S., Szepessy, A., Tempone, R., Zouraris, G.E.: Convergence rates for adaptive weak approximation of stochastic differential equations. Stoch. Anal. Appl. 23, 511–558 (2005)
Oberle, H.J., Pesch, H.J.: Numerical treatment of delay differential equations by Hermit interpolation. Numerische Mathematik 37, 235–255 (1981)
Rössler, A.: Runge-Kutta Metods for the numerical Solution of Stochastic Differential Equation, Ph.D. Thesis. Darmastadt University of Technology, Shaker Verlag, Aachen (2003)
Szepessy, A., Tempone, R., Zouraris, G.E.: Adaptive weak approximation of stochastic differential equations. Commun. Pure Appl. Math. 54, 1169–1214 (2001)
Talay, D., Tubaro, L.: Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8(4), 483–509 (1990)
Thompson, S: Stepsize control for delay differential equations using continuously imbedded Runge-Kutta methods of Sarafyan. J. Comput. Appl. Math. 31, 267–275 (1990)
Tian, T., Burrage, K., Burrage, P.M., Carletti, M.: Stochastic delay differential equations for genetic regulatory networks. J. Comput. Appl. Math. 205, 696–707 (2007)
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Akhtari, B., Babolian, E. & Foroush Bastani, A. An adaptive weak continuous Euler-Maruyama method for stochastic delay differential equations. Numer Algor 69, 29–57 (2015). https://doi.org/10.1007/s11075-014-9880-6
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DOI: https://doi.org/10.1007/s11075-014-9880-6