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Original Articles

Study on split-step Rosenbrock type method for stiff stochastic differential systems

, &
Pages 818-836 | Received 08 Aug 2018, Accepted 26 Feb 2019, Published online: 20 Mar 2019

References

  • Sk.S. Ahmad, N.C. Parida, and S. Raha, The fully implicit stochastic-α method for stiff stochastic differential equations, J. Comput. Phys. 228 (2009), pp. 8263–8282.
  • S. Amiri and S.M. Hosseini, A class of balanced stochastic Runge–Kutta methods for stiff SDE systems, Numer. Algorithms 69 (2015), pp. 531–552.
  • S. Amiri and S.M. Hosseini, A class of weak second order split-drift stochastic Runge–Kutta schemes for stiff SDE systems, J. Comput. Appl. Math. 275 (2015), pp. 27–43.
  • S. Amiri and S.M. Hosseini, Stochastic Runge–Kutta Rosenbrock type methods for SDE systems, Appl. Numer. Math. 115 (2017), pp. 1–15.
  • S. Amiri and A. Rößler, Split-step double balanced approximation methods for stiff stochastic differential equations, Int. J. Comput. Math. (2018). Available at https://doi.org/10.1080/00207160.2018.1480761.
  • J.A.D. Appleby, M. Guzowska, C. Kelly, and A. Rodkina, Preserving positivity in solutions of discretised stochastic differential equations, Appl. Math. Comput. 217 (2010), pp. 763–774.
  • R. Bellman, Stochastic transformations and functional equations, Proc. Sympos. Appl. Math. 16 (1964), pp. 171–177.
  • G. Bonanno, D. Valenti, and B. Spagnolo, Role of noise in a market model with stochastic volatility, Eur. Phys. J. B 53 (2006), pp. 405–409.
  • E. Buckwar and C. Kelly, Towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations, SIAM J. Numer. Anal. 48(1) (2010), pp. 298–321.
  • E. Buckwar and T. Sickenberger, A comparative linear mean-square stability analysis of Maruyama-and Milstein-type methods, Math. Comput. Simul. 81 (2011), pp. 1110–1127.
  • E. Buckwar and T. Sickenberger, A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems, Appl. Numer. Math. 62 (2012), pp. 842–859.
  • K. Burrage and T. Tian, The composite Euler method for stiff stochastic differential equations, J. Comput. Appl. Math. 131 (2000), pp. 407–426.
  • J. Cox, J. Ingersoll, and S. Ross, An intertemporal general equilibrium model of asset prices, Econometrica 53(2) (1985), pp. 363–384.
  • N. Dalal, D. Greenhalgh, and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl. 341 (2008), pp. 1084–1101.
  • S. Dereich, A. Neuenkirch, and L. Szpruch, An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process, Proc. R. Soc. A 468 (2012), pp. 1105–1115.
  • X. Ding, Q. Ma, and L. Zhang, Convergence and stability of the split-step θ-method for stochastic differential equations, Comput. Math. Appl. 60(5) (2010), pp. 1310–1321.
  • A.S. Fatemion Aghda, S.M. Hosseini, and M. Tahmasebi, Analysis of non-negativity and convergence of solution of the balanced implicit method for the delay Cox–Ingersoll–Ross model, Appl. Numer. Math. 118 (2017), pp. 249–265.
  • Q. Guo, H. Li, and Y. Zhu, The improved split-step θ methods for stochastic differential equation, Math. Methods Appl. Sci. 37(15) (2014), pp. 2245–2256.
  • A. Haghighi and S.M. Hosseini, A class of split-step balanced methods for stiff stochastic differential equations, Numer. Algor. 61 (2012), pp. 141–162.
  • A. Haghighi, S.M. Hosseini, and A. Rößler, Diagonally drift-implicit Runge–Kutta methods of strong order one for stiff stochastic differential systems, J. Comput. Appl. Math. 293 (2016), pp. 82–93.
  • N. Halidias, Constructing positivity preserving numerical schemes for the two-factor CIR model, Monte Carlo Methods Appl. 21(4) (2015), pp. 313–323.
  • N. Halidias and I.S. Stamatiou, On the numerical solution of some non-linear stochastic differential equations using the semi-discrete method, Comput. Methods Appl. Math. 16(1) (2016), pp. 105–132.
  • D.J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal. 38 (2000), pp. 753–769.
  • D.J. Higham, X. Mao, and A.M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40(3) (2003), pp. 1041–1063.
  • D.J. Higham, X. Mao, and L. Szpruch, Convergence, non-negativity and stability of a new Milstein scheme with applications to finance, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), pp. 2083–2100.
  • C. Kahl, M. Günther, and T. Rosberg, Structure preserving stochastic integration schemes in interest rate derivative modeling, Appl. Numer. Math. 58 (2008), pp. 284–295.
  • R. Khasminskii, Stochastic Stability of Differential Equations, Vol. 66, Springer-Verlag, Berlin, 2012.
  • P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, Springer–Verlag, Berlin, 1999.
  • J.-C. Li and D.-C. Mei, The influences of delay time on the stability of a market model with stochastic volatility, Phys. A 392 (2013), pp. 763–772.
  • X. Mao, Stochastic Differential Equations and their Applications, Horwood Publishing, Chichester, 1997.
  • G.N. Milstein and M.V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer–Verlag, Berlin, 2004.
  • G.N. Milstein, E. Platen, and H. Schurz, Balanced implicit methods for stiff stochastic systems, SIAM J. Numer. Anal. 35 (1998), pp. 1010–1019.
  • E. Moro and H. Schurz, Boundary preserving semianalytic numerical algorithms for stochastic differential equations, SIAM J. Sci. Comput. 29 (2007), pp. 1525–1549.
  • H. Qian, Counting the Floating Point Operations (FLOPS), MATLAB Central File Exchange, No. 50608, Ver. 1.0, Retrieved June 30, 2015.
  • V. Reshniak, A.Q.M. Khaliq, D.A. Voss, and G. Zhang, Split-step Milstein methods for multi-channel stiff stochastic differential systems, Appl. Numer. Math. 89 (2015), pp. 1–23.
  • A. Rodkina and H. Schurz, On positivity and boundedness of solutions of nonlinear stochastic differential equations, Discrete Contin. Dyn. Syst. 2009 (2009), pp. 640–649.
  • Y. Saito and T. Mitsui, Mean-square stability of numerical schemes for stochastic differential systems, Vietnam J. Math. 30 (2002), pp. 551–560.
  • H. Schurz, Numerical regularization for SDEs: Construction of nonnegative solutions, Dyn. Syst. Appl. 5 (1996), pp. 323–352.
  • S. Singh and S. Raha, Five-stage Milstein methods for SDEs, Int. J. Comput. Math. 89 (2012), pp. 760–779.
  • J. Tan, H. Yang, W. Men, and Y. Guo, Construction of positivity preserving numerical method for stochastic age-dependent population equations, Appl. Math. Comput. 293 (2017), pp. 57–64.
  • T.H. Tian and K. Burrage, Implicit Taylor methods for stiff stochastic differential equations, Appl. Numer. Math. 38 (2001), pp. 167–185.
  • D. Valenti, B. Spagnolo, and G. Bonanno, Hitting time distributions in financial markets, Phys. A 382 (2007), pp. 311–320.
  • D.A. Voss and A.Q.M. Khaliq, Split-step Adams–Moulton Milstein methods for systems of stiff stochastic differential equations, Int. J. Comput. Math. 92 (2015), pp. 995–1011.
  • P. Wang, A-stable Runge–Kutta methods for stiff stochastic differential equations with multiplicative noise, Comput. Appl. Math. 34 (2015), pp. 773–792.
  • P. Wang and Z. Liu, Split-step backward balanced Milstein methods for stiff stochastic differential systems, Appl. Numer. Math. 59 (2009), pp. 1198–1213.
  • X. Wang, S. Gan, and D. Wang, A family of fully implicit Milstein methods for stiff stochastic differential equations with multiplicative noise, BIT Numer. Math. 52 (2012), pp. 741–772.
  • Z. Yin and S. Gan, An error corrected Euler–Maruyama method for stiff stochastic differential equations, Appl. Math. Comput. 256(1) (2015), pp. 630–641.

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