350
Views
20
CrossRef citations to date
0
Altmetric
Original Articles

Multi-Step Maruyama Methods for Stochastic Delay Differential Equations

&
Pages 933-959 | Received 01 Jan 2006, Accepted 13 Nov 2006, Published online: 30 Aug 2007

REFERENCES

  • Appleby , J.A.D. , and Buckwar , E. 2005 . Noise induced oscillation in solutions of stochastic delay differential equations . Dyn. Sys. App. 14 ( 2 ): 175 – 196 .
  • Baker , C.T.H. , and Buckwar , E. 2000 . Numerical analysis of explicit one-step methods for stochastic delay differential equations . LMS J. Comput. Math. 3 : 315 – 335 .
  • Bellen , A. , and Zennaro , M. 2003. Numerical Methods for Delay Differential Equations . Oxford University Press.
  • Beretta , E. , Kolmanovskii , V.B. , and Shaikhet , L. 1998 . Stability of epidemic model with time delays influenced by stochastic perturbations . Math. Comput. Simulation 45 ( 3–4 ): 269 – 277 .
  • Beuter , A. , and Vasilakos , K. 1993 . Effects of noise on a delayed visual feedback system . J. Theor. Biology 165 : 389 – 407 .
  • Bocharov , G.A. , and Rihan , F.A. 2000 . Numerical modelling in bio sciences using delay differential equations . J. Comput. Appl. Math. 125 ( 1–2 ): 183 – 199 .
  • Buckwar , E. 2006 . One-step approximations for stochastic functional differential equations . Appl. Num. Math. 56 ( 5 ): 667 – 681 .
  • Buckwar , E. , and Winkler , R. 2006 . Multi-step methods for SDEs and their application to problems with small noise . SIAM J. Numer. Anal. 44 ( 2 ): 779 – 803 .
  • Buckwar , E. , and Winkler , R. 2004 . On two-step schemes for SDEs with small noise . Proc. Appl. Math. Mech. 4 : 15 – 18 .
  • Buckwar , E. , and Winkler , R. Multi-Step Maruyama Methods for Stochastic Delay Differential Equations . Humboldt–Universität zu Berlin, Institut für Mathematik, Preprint 2004-15 .
  • Chang , M.-H. , and Youree , R.K. 1999 . The European option with hereditary price structures: basic theory . Appl. Math. Comp. 102 : 279 – 296 .
  • Eurich , C.W. , and Milton , J.G. 1996 . Noise-induced transitions in human postural sway . Phys. Rev. E 54 ( 6 ): 6681 – 6684 .
  • Fischer , M. , and Imkeller , P. 2004 . A Two-State Model for Noise Induced Resonance in Bistable Systems with Delay . Preprint .
  • Garcia-Ojalvo , J. , and Roy , R. 1996 . Noise amplification in a stochastic Ikeda model . Phys. Lett. A 224 : 51 – 56 .
  • Gautschi , W. 1997 . Numerical Analysis. An Introduction . Birkhäuser , Boston .
  • Grigoriu , M. 1997 . Control of time delay linear systems with Gaussian white noise . Prob. Eng. Mech. 12 ( 2 ): 89 – 96 .
  • Hairer , E. , Nøtt , S.P. , and Wanner , G. 1993 . Solving Ordinary Differential Equations. I: Nonstiff Problems. , 2. Rev. ed. Springer-Verlag , Berlin .
  • Hale , J. , and Verduyn Lunel , S.M. 1993 . Introduction to Functional Differential Equations . Springer-Verlag , Berlin .
  • Hobson , D.G. , and Rogers , L.C.G. 1998 . Complete models with stochastic volatility . Math. Finance 8 ( 1 ): 27 – 48 .
  • Hu , Y. , Mohammed , S.-E.A. , and Yan , F. 2004 . Discrete-time approximations of stochastic delay differential equations: The Milstein scheme . Annals of Probability 32 ( 1A ): 265 – 314 .
  • Jiang , C. , Troesch , A.W. , and Shaw , S.W. 2000 . Capsize criteria for ship models with memory-dependent hydrodynamics and random excitation . Phil. Trans. R. Soc. Lond. A 358 : 1761 – 1791 .
  • Karatzas , I. , and Shreve , S.E. 1991 . Brownian Motion and Stochastic Calculus . Springer-Verlag , New York .
  • Kim , S. , Park , S.H. , and Ryu , C.S. 1997 . Multistability in coupled oscillator systems with time delay . Phys. Rev. Lett. 79 ( 15 ): 2911 – 2914 .
  • Kloeden , P.E. , and Platen , E. 1992 . Numerical Solution of Stochastic Differential Equations . Springer-Verlag , Berlin .
  • Kolmanovskii˘ , V.B. , and Myshkis , A. 1999. Introduction to the Theory and Applications of Functional Differential Equations . Kluwer Academic Publishers , Dordrecht.
  • Longtin , A. , Milton , J.G. , Bos , J. , and Mackey , M.C. 1990 . Noise and critical behavior of the pupil light reflex at oscillation onset . Phys. Rev. A 41 : 6992 – 7005 .
  • Longtin , A. 1991 . Noise-induced transitions at a Hopf bifurcation in a first order delay-differential equation . Phys. Rev. A 44 : 4801 – 4813 .
  • Mao , X. 1997 . Stochastic Differential Equations and Their Applications . Horwood Publishing Limited , Chichester .
  • Masoller , C. 2002 . Noise-induced resonance in delayed feedback systems . Phys. Rev. Lett. 88 ( 3 ): 034102-1 – 034102-4 .
  • Masoller , C. 2002 . Numerical investigation of noise-induced resonance in a semiconductor laser with optical feedback . Physica D 168–169 : 171 – 176 .
  • Masoller , C. 2003 . Distribution of residence times of time-delayed bistable systems driven by noise . Phys. Rev. Lett. 90 ( 2 ): 020601-1 – 020601-4 .
  • Milstein , G.N. 1995 . Numerical Integration of Stochastic Differential Equations (Translation from the Russian original of 1988) . Kluwer .
  • Milstein , G.N. , and Tretyakov , M.V. 2004 . Stochastic Numerics for Mathematical Physics . Springer-Verlag , Berlin .
  • Mohammed , S.-E.A. 1984 . Stochastic Functional Differential Equations . Pitman , Boston .
  • Nualart , D. 1995 . The Malliavin Calculus and Related Topics . Springer-Verlag , Berlin .
  • Nualart , D. , and Pardoux , E. 1988 . Stochastic calculus with anticipating integrands . Probab. Theory Relat. Fields 78 ( 4 ): 535 – 581 .
  • Peterka , R.J. 2000 . Postural control model interpretation of stabilogram diffusion analysis . Biol Cybern. 82 ( 4 ): 335 – 343 .
  • Plato , R. 2000 . Numerische Mathematik kompakt. Grundlagenwissen für Studium und Praxis . Vieweg , Braunschweig .
  • Römisch , W. , and Winkler , R. 2003 . Stochastic DAEs in circuit simulation . Modeling, Simulation and Optimization of Integrated Circuits . Antreich , K. , Bulirsch , R. , Gilg , A. , and Rentrop , P. (eds.), Birkhäuser , Basel , pp. 303 – 318 .
  • Römisch , W. , and Winkler , R. 2006 . Stepsize control for mean-square numerical methods for SDEs with small noise . SIAM J. Sci. Comp. 28 ( 2 ): 604 – 625 .
  • Shiryaev , A.N. 1996 . Probability. , (Second Ed.) , Springer-Verlag , Berlin .
  • Sichermann , W. , and Kreuzer , E. 2003 . Capsize risk assessment and designing against capsize . Proc. Appl. Math. Mech. 3 ( 1 ): 501 – 502 .
  • Tapaswi , P.K. , and Mukhopadhyay , A. 1999 . Effects of environmental fluctuation on plankton allelopathy . J. Math. Biol. 39 ( 1 ): 39 – 58 .
  • Tsimring , L.S. , and Pikovsky , A. 2001 . Noise-induced dynamics in bistable systems with delay . Phys. Rev. Lett. 87 ( 25 ): 250602-1 – 250602-4 .
  • Winkler , R. 2003 . Stochastic differential algebraic equations of index 1 and applications in circuit simulation . J. Comput. Appl. Math. 157 ( 2 ): 477 – 505 .

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.