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Stochastic Analysis for Obtuse Random Walks

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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.

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Acknowledgments

We would like to thank an anonymous referee for helpful comments and suggestions.

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Authors and Affiliations

  1. Département de Mathématiques de Besançon, UFC, 16 Route de Gray, 25 030 , Besançon Cedex, France

    Uwe Franz

  2. IPEST, Université de Carthage, 2078 , La Marsa, Tunisie

    Tarek Hamdi

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  1. Uwe Franz
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  2. Tarek Hamdi
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Correspondence to Tarek Hamdi.

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Franz, U., Hamdi, T. Stochastic Analysis for Obtuse Random Walks. J Theor Probab 28, 619–649 (2015). https://doi.org/10.1007/s10959-013-0522-z

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  • DOI: https://doi.org/10.1007/s10959-013-0522-z

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