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Strong discrete time approximation of Stochastic Differential Equations with Time Delay

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  • Küchler, U.
  • Platen, E.

Abstract

The paper introduces an approach for the derivation of discrete time approximations for solutions of stochastic differential equations (SDEs) with time delay. The suggested approximations converge in a strong sense. Furthermore, explicit solutions for linear stochastic delay equations are given.
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Suggested Citation

  • Küchler, U. & Platen, E., 1999. "Strong discrete time approximation of Stochastic Differential Equations with Time Delay," SFB 373 Discussion Papers 1999,25, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    • Handle: RePEc:zbw:sfb373:199925
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    1. Eckhard Platen, 1999. "An Introduction to Numerical Methods for Stochastic Differential Equations," Research Paper Series 6, Quantitative Finance Research Centre, University of Technology, Sydney.
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    Cited by:

    1. Mao, Wei & Hu, Liangjian & Mao, Xuerong, 2015. "The existence and asymptotic estimations of solutions to stochastic pantograph equations with diffusion and Lévy jumps," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 883-896.
    2. Yu, Wenwu & Cao, Jinde, 2007. "Synchronization control of stochastic delayed neural networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 373(C), pages 252-260.
    3. Küchler, Uwe & Platen, Eckhard, 2002. "Weak discrete time approximation of stochastic differential equations with time delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 59(6), pages 497-507.
    4. Buckwar, Evelyn & Shardlow, Tony, 2001. "Weak approximation of stochastic differential delay equations," SFB 373 Discussion Papers 2001,88, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    5. Wang, Zhen & Li, Xiong & Lei, Jinzhi, 2014. "Moment boundedness of linear stochastic delay differential equations with distributed delay," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 586-612.
    6. Yuan, Chenggui & Mao, Xuerong, 2004. "Convergence of the Euler–Maruyama method for stochastic differential equations with Markovian switching," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 64(2), pages 223-235.
    7. Mao, Wei & Zhu, Quanxin & Mao, Xuerong, 2015. "Existence, uniqueness and almost surely asymptotic estimations of the solutions to neutral stochastic functional differential equations driven by pure jumps," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 252-265.
    8. Uwe Küchler & Michael Sørensen, 2010. "A simple estimator for discrete-time samples from affine stochastic delay differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 13(2), pages 125-132, June.
    9. Becker, Christoph & Schmidt, Wolfgang M., 2013. "Stressing correlations and volatilities — A consistent modeling approach," Journal of Empirical Finance, Elsevier, vol. 21(C), pages 174-194.
    10. Hu, Rong, 2020. "Almost sure exponential stability of the Milstein-type schemes for stochastic delay differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    11. Marco Ferrante & Elisabetta Ferraris & Carles Rovira, 2016. "On a stochastic epidemic SEIHR model and its diffusion approximation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(3), pages 482-502, September.
    12. Eckhard Platen, 2020. "Stochastic Modelling of the COVID-19 Epidemic," Research Paper Series 409, Quantitative Finance Research Centre, University of Technology, Sydney.
    13. Uwe Küchler & Eckhard Platen, 2007. "Time Delay and Noise Explaining Cyclical Fluctuations in Prices of Commodities," Research Paper Series 195, Quantitative Finance Research Centre, University of Technology, Sydney.
    14. Xiaopeng Xi & Donghua Zhou, 2022. "Prognostics of fractional degradation processes with state-dependent delay," Journal of Risk and Reliability, , vol. 236(1), pages 114-124, February.
    15. Mahmoudi, Fatemeh & Tahmasebi, Mahdieh, 2022. "The convergence of a numerical scheme for additive fractional stochastic delay equations with H>12," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 191(C), pages 219-231.

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