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On the Strong Approximation of Jump-Diffusion Processes

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Abstract

In financial modelling, filtering and other areas the underlying dynamics are often specified via stochastic differential equations (SDEs) of jump-diffusion type. The class of jump-diffusion SDEs that admits explicit solutions is rather limited. Consequently, there is a need for the systematic use of discrete time approximations in corresponding simulations. This paper presents a survey and new results on strong numerical schemes for SDEs of jump-diffusion type. These are relevant for scenario analysis, filtering and hedge simulation in finance. It provides a convergence theorem for the construction of strong approximations of any given order of convergence for SDEs driven by Wiener processes and Poisson random measures. The paper covers also derivative free, drift-implicit and jump adapted strong approximations. For the commutative case particular schemes are obtained. Finally, a numerical study on the accuracy of several strong schemes is presented.

Suggested Citation

  • Nicola Bruti-Liberati & Eckhard Platen, 2005. "On the Strong Approximation of Jump-Diffusion Processes," Research Paper Series 157, Quantitative Finance Research Centre, University of Technology, Sydney.
  • Handle: RePEc:uts:rpaper:157
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    File URL: http://www.qfrc.uts.edu.au/research/research_papers/rp157.pdf
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    References listed on IDEAS

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    1. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    2. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    3. Tomas Björk & Yuri Kabanov & Wolfgang Runggaldier, 1997. "Bond Market Structure in the Presence of Marked Point Processes," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 211-239.
    4. Carl Chiarella & Christina Sklibosios, 2003. "A Class of Jump-Diffusion Bond Pricing Models within the HJM Framework," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 10(2), pages 87-127, September.
    5. Philippe Jorion, 1988. "On Jump Processes in the Foreign Exchange and Stock Markets," Review of Financial Studies, Society for Financial Studies, vol. 1(4), pages 427-445.
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    Citations

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    Cited by:

    1. Nicola Bruti-Liberati & Eckhard Platen, 2006. "On Weak Predictor-Corrector Schemes for Jump-Diffusion Processes in Finance," Research Paper Series 179, Quantitative Finance Research Centre, University of Technology, Sydney.
    2. Nicola Bruti-Liberati & Eckhard Platen, 2005. "On the Strong Approximation of Pure Jump Processes," Research Paper Series 164, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Pellegrini, Clément, 2010. "Existence, uniqueness and approximation of the jump-type stochastic Schrodinger equation for two-level systems," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1722-1747, August.
    4. Nicola Bruti-Liberati & Eckhard Platen, 2007. "Approximation of jump diffusions in finance and economics," Computational Economics, Springer;Society for Computational Economics, vol. 29(3), pages 283-312, May.
    5. Szimayer, Alex & Maller, Ross A., 2007. "Finite approximation schemes for Lévy processes, and their application to optimal stopping problems," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1422-1447, October.
    6. Yuan Xia, 2011. "Multilevel Monte Carlo method for jump-diffusion SDEs," Papers 1106.4730, arXiv.org.

    More about this item

    Keywords

    jump-diffusion processes; stochastic Taylor expansion; discrete time approximation; simulation; strong convergence;

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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