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Original Articles

Generalized two-step Milstein methods for stochastic differential equations

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Pages 1363-1379 | Received 17 Sep 2018, Accepted 13 Apr 2019, Published online: 21 May 2019

References

  • L. Arnold, Stochastic Differential Equations: Theory and Applications, John Wiley & Sons, New York, 1974.
  • R.H. Bokor, On two-step methods for stochastic differential equations, Acta Cybernet. 13(2) (1998), pp. 197–207.
  • R.H. Bokor, The strong convergence and numerical stability of multistep approximations of solutions of stochastic ordinary differential equations, Stoch. Anal. Appl. 25 (2007), pp. 1–38. doi: 10.1081/SAP-200056694
  • L. Brugnano, K. Burrage, and P.M. Burrage, Adams-type methods for the numerical solution of stochastic ordinary differential equations, BIT Numer. Math. 40(3) (1998), pp. 451–470. doi: 10.1023/A:1022363612387
  • 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, Improved linear multistep methods for stochastic ordinary differential equations, J. Comput. Appl. Math. 205 (2007), pp. 912–922. doi: 10.1016/j.cam.2006.03.038
  • E. Buckwar, R.H. 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
  • G. Denk and S. Schäffler, Adams methods for the efficient solution of stochastic differential equations with additive noise, Computing 59(2) (1997), pp. 153–161. doi: 10.1007/BF02684477
  • S. Elaydi, An Introduction to Difference Equations, Springer, New York, 2005.
  • B.D. Ewald and R. Témam, Numerical analysis of stochastic schemes in geophysics, SIAM J. Numer. Anal. 42(6) (2005), pp. 2257–2276. doi: 10.1137/S0036142902418333
  • E. Jury, Theory and Applications of the Z-transform, Willy, New York, 1964.
  • P.E. Kloeden and E. Platen, Higher-order implicit strong numerical schemes for stochastic differential equations, J. Stat. Phys. 66(1–2) (1992), pp. 283–314. doi: 10.1007/BF01060070
  • J.D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley & Sons, Chichester, 1991.
  • G.N. Milstein, Numerical Integration of Stochastic Differential Equations, Springer, Dordrecht, 1995.
  • G.N. Milstein and M.V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, Heidelberg, 2004.
  • E. Platen and P.E. Kloeden, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, Heidelberg, 1992.
  • Q. Ren and H. Tian, Generalized two-step Maruyama methods for stochastic differential equations, Appl. Math. Comput. 332 (2018), pp. 48–57.
  • Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal. 33(6) (1996), pp. 2254–2267. doi: 10.1137/S0036142992228409
  • 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, Asymptotic mean-square stability of two-step Maruyama schemes for stochastic differential equations, J. Comput. Appl. Math. 260(2) (2014), pp. 337–348. doi: 10.1016/j.cam.2013.10.002
  • A. Tocino and M.J. Senosiain, Two-step Milstein schemes for stochastic differential equations, Numer. Algorithms 69(3) (2015), pp. 643–665. doi: 10.1007/s11075-014-9918-9

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