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On the Numerical Stability of Simulation Methods for SDES

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Abstract

When simulating discrete time approximations of solutions of stochastic differential equations (SDEs), numerical stability is clearly more important than numerical efficiency or some higher order of convergence. Discrete time approximations of solutions of SDEs are widely used in simulations in finance and other areas of application. The stability criterion presented is designed to handle both scenario simulation and Monte Carlo simulation, that is, strong and weak simulation methods. The symmetric predictor-corrector Euler method is shown to have the potential to overcome some of the numerical instabilities that may be experienced when using the explicit Euler method. This is of particular importance in finance, where martingale dynamics arise for solutions of SDEs and diffusion coefficients are often of multiplicative type. Stability regions for a range of schemes are visualized and discussed. For Monte Carlo simulation it turns out that schemes, which have implicitness in both the drift and the diffusion terms, exhibit the largest stability regions. It will be shown that refining the time step size in a Monte Carlo simulation can lead to numerical instabilities.

Suggested Citation

  • 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.
    • Handle: RePEc:uts:rpaper:234
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    File URL: https://www.uts.edu.au/sites/default/files/qfr-archive-02/QFR-rp234.pdf
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    References listed on IDEAS

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    1. N. Hofmann & Eckhard Platen, 1994. "Stability of weak numerical schemes for stochastic differential equations," Published Paper Series 1994-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    2. Yoshihiro Saito & Taketomo Mitsui, 1993. "Simulation of stochastic differential equations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 45(3), pages 419-432, September.
    3. Eckhard Platen, 2006. "A Benchmark Approach To Finance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 131-151, January.
    4. 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.
    5. 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.
    6. G. N. Milstein & Eckhard Platen & H. Schurz, 1998. "Balanced Implicit Methods for Stiff Stochastic Systems," Published Paper Series 1998-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    7. Platen, Eckhard, 1995. "On weak implicit and predictor-corrector methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 38(1), pages 69-76.
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    Cited by:

    1. 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.
    2. 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.
    3. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 23, July-Dece.
    4. Renata Rendek, 2013. "Modeling Diversified Equity Indices," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 4-2013.

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    Keywords

    stochastic differential equations; scenario simulation; Monte Carlo simulation; numerical stability; predictor-corrector methods; implicit methods;
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