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Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients

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  • Chan, K. S.
  • Stramer, O.

Abstract

We prove that, under appropriate conditions, the sequence of approximate solutions constructed according to the Euler scheme converges weakly to the (unique) solution of a stochastic differential equation with discontinuous coefficients. We also obtain a sufficient condition for the existence of a solution to a stochastic differential equation with discontinuous coefficients. These results are then applied to justify the technique of simulating continuous-time threshold autoregressive moving-average processes via the Euler scheme.

Suggested Citation

  • Chan, K. S. & Stramer, O., 1998. "Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients," Stochastic Processes and their Applications, Elsevier, vol. 76(1), pages 33-44, August.
    • Handle: RePEc:eee:spapps:v:76:y:1998:i:1:p:33-44
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    References listed on IDEAS

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    1. Brockwell, P. J. & Hyndman, R. J., 1992. "On continuous-time threshold autoregression," International Journal of Forecasting, Elsevier, vol. 8(2), pages 157-173, October.
    2. Darrell Duffie & Philip Protter, 1992. "From Discrete‐ to Continuous‐Time Finance: Weak Convergence of the Financial Gain Process1," Mathematical Finance, Wiley Blackwell, vol. 2(1), pages 1-15, January.
    3. O. Stramer, 1996. "On The Approximation Of Moments For Continuous Time Threshold Arma Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 17(2), pages 189-202, March.
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    Cited by:

    1. Mark Podolskij & Bezirgen Veliyev & Nakahiro Yoshida, 2018. "Edgeworth expansion for Euler approximation of continuous diffusion processes," CREATES Research Papers 2018-28, Department of Economics and Business Economics, Aarhus University.
    2. Yao Tung Huang & Qingshuo Song & Harry Zheng, 2015. "Weak Convergence of Path-Dependent SDEs in Basket CDS Pricing with Contagion Risk," Papers 1506.00082, arXiv.org, revised May 2016.
    3. Lingohr, Daniel & Müller, Gernot, 2019. "Stochastic modeling of intraday photovoltaic power generation," Energy Economics, Elsevier, vol. 81(C), pages 175-186.
    4. Antoine Lejay & Paolo Pigato, 2017. "A threshold model for local volatility: evidence of leverage and mean reversion effects on historical data," Working Papers hal-01669082, HAL.
    5. Hiderah Kamal, 2020. "Approximation of Euler–Maruyama for one-dimensional stochastic differential equations involving the maximum process," Monte Carlo Methods and Applications, De Gruyter, vol. 26(1), pages 33-47, March.
    6. Antoine Lejay & Paolo Pigato, 2019. "A Threshold Model For Local Volatility: Evidence Of Leverage And Mean Reversion Effects On Historical Data," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(04), pages 1-24, June.
    7. Ngo, Hoang-Long & Taguchi, Dai, 2017. "Strong convergence for the Euler–Maruyama approximation of stochastic differential equations with discontinuous coefficients," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 55-63.

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