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Stochastic Analysis for Obtuse Random Walks

Author

Listed:
  • Uwe Franz

    (UFC)

  • Tarek Hamdi

    (IPEST, Université de Carthage)

Abstract

We present a construction of the basic operators of stochastic analysis (gradient and divergence) for a class of discrete-time normal martingales called obtuse random walks. The approach is based on the chaos representation property and discrete multiple stochastic integrals. We show that these operators satisfy similar identities as in the case of the Bernoulli random walks. We prove a Clark–Ocone-type predictable representation formula, obtain two covariance identities and derive a deviation inequality. We close the exposition by an application to option hedging in discrete time.

Suggested Citation

  • Uwe Franz & Tarek Hamdi, 2015. "Stochastic Analysis for Obtuse Random Walks," Journal of Theoretical Probability, Springer, vol. 28(2), pages 619-649, June.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:2:d:10.1007_s10959-013-0522-z
    DOI: 10.1007/s10959-013-0522-z
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    References listed on IDEAS

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    1. Stéphane Attal & Ameur Dhahri, 2010. "Repeated Quantum Interactions and Unitary Random Walks," Journal of Theoretical Probability, Springer, vol. 23(2), pages 345-361, June.
    2. J. Jacod & A.N. Shiryaev, 1998. "Local martingales and the fundamental asset pricing theorems in the discrete-time case," Finance and Stochastics, Springer, vol. 2(3), pages 259-273.
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