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Higher-order implicit strong numerical schemes for stochastic differential equations

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Abstract

Higher-order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed. These are based on a stochastic Taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero. The regions of absolute stability of these implicit and related explicit methods are also examined.

Suggested Citation

  • P. E. Kloeden & Eckhard Platen, 1992. "Higher-order implicit strong numerical schemes for stochastic differential equations," Published Paper Series 1992-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    • Handle: RePEc:uts:ppaper:1992-1
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    File URL: https://link.springer.com/article/10.1007/BF01060070
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    1. Lay Harold A. & Colgin Zane & Reshniak Viktor & Khaliq Abdul Q. M., 2018. "On the implementation of multilevel Monte Carlo simulation of the stochastic volatility and interest rate model using multi-GPU clusters," Monte Carlo Methods and Applications, De Gruyter, vol. 24(4), pages 309-321, December.
    2. Quansah, Emmanuel & Parshad, Rana D. & Mondal, Sumona, 2017. "Cold induced mortality of the Burmese Python: An explanation via stochastic analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 467(C), pages 356-364.
    3. Patrick Assonken & Gangaram Ladde, 2017. "Simulation and Calibration of Options Prices under a Levy-Type Stochastic Dynamic and Semi Markov Market Switching Regimes Processes," Applied Economics and Finance, Redfame publishing, vol. 4(1), pages 93-126, January.
    4. Eckhard Platen & Lei Shi, 2008. "On the Numerical Stability of Simulation Methods for SDES," Research Paper Series 234, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Eckhard Platen & Renata Rendek, 2009. "Exact Scenario Simulation for Selected Multi-dimensional Stochastic Processes," Research Paper Series 259, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. Yin, Zhengwei & Gan, Siqing, 2015. "An error corrected Euler–Maruyama method for stiff stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 630-641.
    7. Rupak Chatterjee & Zhenyu Cui & Jiacheng Fan & Mingzhe Liu, 2018. "An efficient and stable method for short maturity Asian options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 38(12), pages 1470-1486, December.
    8. Mahmoud A. Eissa & Boping Tian, 2017. "Lobatto-Milstein Numerical Method in Application of Uncertainty Investment of Solar Power Projects," Energies, MDPI, vol. 10(1), pages 1-19, January.
    9. Eckhard Platen & Renata Rendek, 2009. "Quasi-exact Approximation of Hidden Markov Chain Filters," Research Paper Series 258, Quantitative Finance Research Centre, University of Technology, Sydney.
    10. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 23, July-Dece.
    11. Chassagneux, Jean-François & Richou, Adrien, 2019. "Rate of convergence for the discrete-time approximation of reflected BSDEs arising in switching problems," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4597-4637.
    12. Mu Niu & Pokman Cheung & Lizhen Lin & Zhenwen Dai & Neil Lawrence & David Dunson, 2019. "Intrinsic Gaussian processes on complex constrained domains," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(3), pages 603-627, July.
    13. Alejandra López-Pérez & Manuel Febrero-Bande & Wencesalo González-Manteiga, 2021. "Parametric Estimation of Diffusion Processes: A Review and Comparative Study," Mathematics, MDPI, vol. 9(8), pages 1-27, April.
    14. Nicola Bruti-Liberati & Eckhard Platen, 2008. "Strong Predictor-Corrector Euler Methods for Stochastic Differential Equations," Research Paper Series 222, Quantitative Finance Research Centre, University of Technology, Sydney.
    15. Ren, Quanwei & Tian, Hongjiong, 2018. "Generalized two-step Maruyama methods for stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 48-57.
    16. Hofmann, Norbert, 1995. "Stability of weak numerical schemes for stochastic differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 38(1), pages 63-68.
    17. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 4-2013, January-A.
    18. Shunwei Zhu & Bo Wang, 2019. "Unified Approach for the Affine and Non-affine Models: An Empirical Analysis on the S&P 500 Volatility Dynamics," Computational Economics, Springer;Society for Computational Economics, vol. 53(4), pages 1421-1442, April.

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