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Quasi-shuffle Algebras in Non-commutative Stochastic Calculus

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Geometry and Invariance in Stochastic Dynamics (RTISD19 2019)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 378))

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Abstract

This chapter is divided into two parts. The first is largely expository and builds on Karandikar’s axiomatisation of Itô calculus for matrix-valued semimartingales. Its aim is to unfold in detail the algebraic structures implied for iterated Itô and Stratonovich integrals. These constructions generalise the classical rules of Chen calculus for deterministic scalar-valued iterated integrals. The second part develops the stochastic analog of what is commonly called chronological calculus in control theory. We obtain in particular a pre-Lie Magnus formula for the logarithm of the Itô stochastic exponential of matrix-valued semimartingales.

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Notes

  1. 1.

    Karandikar’s axioms appeared in a 1982 paper [53]. They were (re-)discovered almost two decades later in a completely different context –the one of Stasheff polytopes– as axioms for dendriform trialgebras [56].

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Acknowledgements

The second author would like to express his gratitude for the warm hospitality he experienced during the Verona meeting, with special thoughts for S. Albeverio and S. Ugolini. This work was supported by the French government, managed by the ANR under the UCA JEDI Investments for the Future project, reference number ANR-15-IDEX-01.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology – NTNU, 7491, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  2. Labo. J.-A. Dieudonné, UMR 7351, CNRS, Université Côte d’Azur, 06108 Nice Cedex 02, Parc Valrose, France

    Frédéric Patras

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  1. Kurusch Ebrahimi-Fard
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Correspondence to Frédéric Patras .

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  1. Department of Mathematics, Università degli Studi di Milano, Milan, Italy

    Stefania Ugolini

  2. Department of Mathematics, Università degli Studi di Milano, Milan, Italy

    Marco Fuhrman

  3. Department of Economics, Università degli Studi dell'Insubria, Varese, Italy

    Elisa Mastrogiacomo

  4. Department of Agricultural and Environmental Sciences, Università degli Studi di Milano, Milan, Italy

    Paola Morando

  5. Mathematic and Informatic Department, Bergische Universität Wuppertal, Wuppertal, Germany

    Barbara Rüdiger

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Ebrahimi-Fard, K., Patras, F. (2021). Quasi-shuffle Algebras in Non-commutative Stochastic Calculus. In: Ugolini, S., Fuhrman, M., Mastrogiacomo, E., Morando, P., Rüdiger, B. (eds) Geometry and Invariance in Stochastic Dynamics. RTISD19 2019. Springer Proceedings in Mathematics & Statistics, vol 378. Springer, Cham. https://doi.org/10.1007/978-3-030-87432-2_6

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