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On the simulation of iterated Itô integrals

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  • Rydén, Tobias
  • Wiktorsson, Magnus

Abstract

We consider algorithms for simulation of iterated Itô integrals with application to simulation of stochastic differential equations. The fact that the iterated Itô integralconditioned on Wi(tn+h)-Wi(tn) and Wj(tn+h)-Wj(tn), has an infinitely divisible distribution utilised for the simultaneous simulation of Iij(tn,tn+h), Wi(tn+h)-Wi(tn) and Wj(tn+h)-Wj(tn). Different simulation methods for the iterated Itô integrals are investigated. We show mean-square convergence rates for approximations of shot-noise type and asymptotic normality of the remainder of the approximations. This together with the fact that the conditional distribution of Iij(tn,tn+h), apart from an additive constant, is a Gaussian variance mixture used to achieve an improved convergence rate. This is done by a coupling method for the remainder of the approximation.

Suggested Citation

  • Rydén, Tobias & Wiktorsson, Magnus, 2001. "On the simulation of iterated Itô integrals," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 151-168, January.
    • Handle: RePEc:eee:spapps:v:91:y:2001:i:1:p:151-168
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    References listed on IDEAS

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    1. P. E. Kloeden & Eckhard Platen & I. W. Wright, 1992. "The approximation of multiple stochastic integrals," Published Paper Series 1992-2, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
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    Cited by:

    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. de la Cruz, H., 2020. "Stabilized explicit methods for the approximation of stochastic systems driven by small additive noises," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    3. Mike Giles & Lukasz Szpruch, 2012. "Multilevel Monte Carlo methods for applications in finance," Papers 1212.1377, arXiv.org.
    4. Malham, Simon J.A. & Wiese, Anke, 2014. "Efficient almost-exact Lévy area sampling," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 50-55.
    5. del Baño Rollin, Sebastian & Ferreiro-Castilla, Albert & Utzet, Frederic, 2010. "On the density of log-spot in the Heston volatility model," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 2037-2063, September.

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