276
Views
6
CrossRef citations to date
0
Altmetric
Original Articles

Double-implicit and split two-step Milstein schemes for stochastic differential equations

, , &
Pages 1987-2011 | Received 01 Feb 2015, Accepted 30 Jul 2015, Published online: 02 Sep 2015

References

  • B.A. Bodo, M.E. Thompson, and T.E. Unny, A review on stochastic differential equations for applications in hydrology, Stochastic Hydrol. Hydraul. 1(2) (1987), pp. 81–100. doi: 10.1007/BF01543805
  • E. Buckwar, R. Horváth-Bokor, and R. Winkler, Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations, BIT Numer. Math. 46(2) (2006), pp. 261–282. doi: 10.1007/s10543-006-0060-5
  • E. Buckwar and R. Winkler, On two-step schemes for SDEs with small noise, PAMM 4(1) (2004), pp. 15–18. doi: 10.1002/pamm.200410004
  • E. Buckwar and R. Winkler, Multistep methods for SDEs and their application to problems with small noise, SIAM J. Numer. Anal. 44(2) (2006), pp. 779–803. doi: 10.1137/040602857
  • E. Buckwar and R. Winkler, Multi-step Maruyama methods for stochastic delay differential equations, Stoch. Anal. Appl. 25(5) (2007), pp. 933–959. doi: 10.1080/07362990701540311
  • Q. Guo, H. Li, and Y. Zhu, The improved split-step θ methods for stochastic differential equation, Math. Methods App. Sci. (2013). doi: 10.1007/978-1-4614-6992-6
  • D.J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev. 43(3) (2001), pp. 525–546. doi: 10.1137/S0036144500378302
  • 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) (2002), pp. 1041–1063. doi: 10.1137/S0036142901389530
  • D.J. Higham, X. Mao, and L. Szpruch, Convergence, non-negativity and stability of a new Milstein scheme with applications to finance, Discret. Contin. Dyn. Syst. Ser. B 18(8) (2013).
  • Y. Hu, S.-E.A. Mohammed, and F. Yan, Discrete-time approximations of stochastic delay equations: the Milstein scheme, Ann. Prob. 32(1A) (2004), pp. 265–314. doi: 10.1214/aop/1078415836
  • C. Huang, Exponential mean square stability of numerical methods for systems of stochastic differential equations, J. Comput. Appl. Math. 236(16) (2012), pp. 4016–4026. doi: 10.1016/j.cam.2012.03.005
  • C. Huang, Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations, J. Comput. Appl. Math. 259 (2014), pp. 77–86. doi: 10.1016/j.cam.2013.03.038
  • P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Vol. 23, Springer, New York, 1992.
  • X. Mao, Stochastic Differential Equations and Applications, Elsevier, Amsterdam, 2007.
  • G. Maruyama, Continuous Markov processes and stochastic equations, Rend. Circolo Math. Palemo 4(1) (1955), pp. 48–90. doi: 10.1007/BF02846028
  • G.N. Mil'shtejn, Approximate integration of stochastic differential equations, Theory Probab. Appl. 19(3) (1975), pp. 557–562. doi: 10.1137/1119062
  • G.N. Milstein and M.V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, 2004.
  • E. Platen and N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Vol. 64, Springer, Berlin, 2010.
  • 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. doi: 10.1016/j.apnum.2014.10.005
  • T. Sickenberger, Mean-square convergence of stochastic multi-step methods with variable step-size, J. Comput. Appl. Math. 212(2) (2008), pp. 300–319. doi: 10.1016/j.cam.2006.12.014
  • A. Tocino and M.J. Senosiain, Two-step Milstein schemes for stochastic differential equations, Numer. Algorithms (2014), pp. 1–23.
  • N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, Vol. 1, Elsevier, Amsterdam, 1992.
  • X. Wang and S. Gan, The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Differ. Equ. Appl. 19(3) (2013), pp. 466–490. doi: 10.1080/10236198.2012.656617
  • 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(3) (2012), pp. 741–772. doi: 10.1007/s10543-012-0370-8
  • C. Yue, C. Huang, and F. Jiang, Strong convergence of split-step theta methods for non-autonomous stochastic differential equations, Int. J. Comput. Math. 91(10) (2014), pp. 2260–2275. doi: 10.1080/00207160.2013.871541
  • E. Zeidler, Nonlinear Functional Analysis and Its Applications, ii/a: Linear Monotone Operators, ii/b: Nonlinear Monotone Operators, Springer, New York, 1990.
  • X. Zong, F. Wu, and C. Huang, Preserving exponential mean square stability and decay rates in two classes of theta approximations of stochastic differential equations, J. Diff. Equ. Appl. 20(7) (2014), pp. 1091–1111. doi: 10.1080/10236198.2014.892934
  • X. Zong, F. Wu, and C. Huang, Theta schemes for SDDEs with non-globally lipschitz continuous coefficients, J. Comput. Appl. Math. 278 (2015), pp. 258–277. doi: 10.1016/j.cam.2014.10.014
  • X. Zong, F. Wu, and G. Xu, Convergence and stability of two classes of theta-Milstein schemes for stochastic differential equations, preprint (2015). Available at arXiv:1501.03695.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.