Abstract
In this paper, we first investigate the stability of two weak second order methods introduced by Debrabant and Rößler (Appl Numer Math 59:582–594, 2009) and Platen (Math Comput Simulation 38:69–76, 1995). We then propose a new weak second order predictor-corrector method, with an improved stability properties, based on the Rößler’s method as the predictor and the implicit method of Platen as the corrector. The stability functions of these methods, applied to a scalar linear test equation with multiplicative noise, are determined and their regions of stability are then compared with the corresponding stability regions of the test equation. Furthermore, we also investigate mean square stability (MS-stability) of these methods applied to a linear Itô 2-dimensional stochastic differential test equation. Numerical examples will be presented to support the theoretical results.
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References
Foroush Bastani, A., Hosseini, S.M.: A new adaptive Runge Kutta method for stochastic differential equations. J. Comput. Appl. Math. 206, 631–644 (2007)
Burrage, K., Burrage, P.M., Mitsui, T.: Numerical solutions of stochastic differential equations- implementation and stability issues. J. Comput. Appl. Math. 125, 171–182 (2000)
Burrage, K., Tian, T.H.: Stiffly accurate Runge Kutta methods for stiff stochastic differential equations. Comput. Phys. Commun. 142, 186–190 (2001)
Debrabant, K., Rößler, A.: Families of efficient second order Runge Kutta methods for the weak approximation of Itô stochastic differential equations. Appl. Numer. Math. 59, 582–594 (2009)
Debrabant, K., Rößler, A.: Diagonally drift-implicit Runge Kutta methods of weak order one and two for Itô SDEs and stability analysis. Appl. Numer. Math. 59, 595–607 (2009)
Hernandez, D.B., Spigler, R.: Convergence and stability of implicit Runge Kutta methods for systems with multiplicative noise. BIT 33(4), 654–669 (1993)
Higham, D.J.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38(3), 753–769 (2000)
Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. Edu. Sect. 43, 525–546 (2001)
Hofmann, N., Platen, E.: Stability of weak numerical schemes for stochastic differential equations. Comput Math. Appl. 28(10–12), 45–57 (1994)
Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations. Applications of Mathematics, vol. 23. Springer, Berlin (1999)
Komori, Y., Mitsui, T.: Stable ROW-type weak scheme for stochastic differential equations. Monte Carlo Methods Appl. 1(4), 279–300 (1995)
Lewis, D.W.: Matrix Theory. World Scientific New Jersey, Singapore (1991)
Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Kluwer, Dordrecht (1995)
Milstein, G.N., Platen, E., Schurz, H.: Balanced implicit methods for stiff stochastic systems. SIAM J. Numer. Anal. 35, 1010–1019 (1998)
Petersen, W.P.: A general implicit splitting for stabilizing numerical simulations of Itô stochastic differential equations. SIAM. J. Numer. Anal. 35, 1439–1451 (1998)
Platen, E.: On weak implicit and predictor-corrector methods. Math. Comput. Simulation 38, 69–76 (1995)
Rößler, A.: Second order Runge Kutta methods for Itô stochastic differential equations. Preprint No. 2479, Technische Universität Darmstadt (2006)
Rößler, A.: Second order Runge Kutta methods for Stratonovich stochastic differential equations. BIT 47(3), 657–680 (2007)
Saito, Y., Mitsui, T.: Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33(6), 2254–2267 (1996)
Saito, Y., Mitsui, T.: Mean-square stability of numerical schemes for stochastic differential systems. In: International Conference on Scientific Computation and Differential Equations, July 29–August 3, 2001. Vancouver, British Columbia, Canada (2001)
Tian, T.H., Burrage, K.: Implicit Taylor methods for stiff stochastic differential equations. Appl. Numer. Math. 38, 167–185 (2001)
Valinejad, A., Hosseini, S.M.: A variable step-size control algorithm for the weak approximations of stochastic differential equations. Numer. Algorithms (2010). doi:10.1007/s11075-010-9363-3
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Haghighi, A., Hosseini, S.M. On the stability of some second order numerical methods for weak approximation of Itô SDEs. Numer Algor 57, 101–124 (2011). https://doi.org/10.1007/s11075-010-9417-6
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DOI: https://doi.org/10.1007/s11075-010-9417-6