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On Non-commutative Stochastic Exponentials

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Numerical Mathematics and Advanced Applications ENUMATH 2017 (ENUMATH 2017)

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Abstract

Using non-commutative shuffle algebra, we outline how the Magnus expansion allows to define explicit stochastic exponentials for matrix-valued continuous semimartingales and Stratonovich integrals.

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References

  1. A. Agrachev, R. Gamkrelidze, Chronological algebras and nonstationary vector fields. J. Sov. Math. 17, 1650–1675 (1981)

    Article  Google Scholar 

  2. M. Arnaudon, Semi-martingales dans les espaces homogènes. Ann. Inst. Henri Poincaré. 29(2), 269–288 (1993)

    MathSciNet  MATH  Google Scholar 

  3. P. Biane, Free Brownian motion, free stochastic calculus and random matrices. Fields Inst. Commun. 12, 1–19 (1997)

    MathSciNet  MATH  Google Scholar 

  4. Ph. Biane, R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space. Probab. Theory Relat. Fields 112(3), 373–409 (1998)

    Article  MathSciNet  Google Scholar 

  5. S. Blanes, F. Casas, J.A. Oteo, J. Ros, Magnus expansion: mathematical study and physical applications. Phys. Rep. 470, 151–238 (2009)

    Article  MathSciNet  Google Scholar 

  6. P. Cartier, Vinberg algebras, Lie groups and combinatorics. Clay Math. Proc. 11, 107–126 (2011)

    MATH  Google Scholar 

  7. C. Curry, K. Ebrahimi-Fard, S.J.A. Malham, A. Wiese, Lévy processes and quasi-shuffle algebras. Stochastics 86(4), 632–642 (2014)

    Article  MathSciNet  Google Scholar 

  8. K. Ebrahimi-Fard, D. Manchon, A Magnus- and Fer-type formula in dendriform algebras. Found. Comput. Math. 9, 295–316 (2009)

    Article  MathSciNet  Google Scholar 

  9. K. Ebrahimi-Fard, S.J.A. Malham, F. Patras, A. Wiese, The exponential Lie series for continuous semimartingales. Proc. R. Soc. A 471, 20150429 (2015)

    Article  MathSciNet  Google Scholar 

  10. K. Ebrahimi-Fard, S.J.A. Malham, F. Patras, A. Wiese, Flows and stochastic Taylor series in Ito calculus. J. Phys. A Math. Theor. 48, 495202 (2015)

    Article  MathSciNet  Google Scholar 

  11. M. Emery, Stabilité des solutions des équations différentielles stochastiques application aux intégrales multiplicatives stochastiques. Z. Wahrscheinlichkeitstheorie verw. Gebiete 41, 241–262 (1978)

    Article  Google Scholar 

  12. M. Hakim-Dowek, D. Lépingle, L’exponentielle stochastique de groupes de Lie. Lec. Notes Math. 1204, 352–374 (1986)

    Article  MathSciNet  Google Scholar 

  13. R.L. Hudson, K.R. Parthasarathy, Quantum Itô’s formula and stochastic evolutions. Commun. Math. Phys. 93(3), 301–323 (1984)

    Article  Google Scholar 

  14. F. Jamshidian, On the combinatorics of iterated stochastic integrals. Stochastics 83(1), 1–15 (2011)

    Article  MathSciNet  Google Scholar 

  15. R.L. Karandikar, Multiplicative decomposition of non-singular matrix valued continuous semimartingales. Ann. Probab. 10, 1088–1091 (1982)

    Article  MathSciNet  Google Scholar 

  16. V. Kargin, On free stochastic differential equations. J. Theor. Probab. 24(3), 821–848 (2011)

    Article  MathSciNet  Google Scholar 

  17. W. Magnus, On the exponential solution of differential equations for a linear operator. Commun. Pure Appl. Math. 7, 649–673 (1954)

    Article  MathSciNet  Google Scholar 

  18. D. Manchon, A short survey on pre-Lie algebras, in Non-commutative Geometry and Physics: Renormalisation, Motives, Index Theory, ed. by A. Carey. E. Schrödinger Institut Lectures in Mathematics and Physics (European Mathematical Society, Helsinki, 2011)

    Google Scholar 

  19. B. Mielnik, J. Plebański, Combinatorial approach to Baker–Campbell–Hausdorff exponents. Ann. Inst. Henri Poincaré A XII, 215–254 (1970)

    Google Scholar 

  20. P.E. Protter, Stochastic Integration and Differential Equations, Version 2.1, 2nd edn. (Springer, Berlin, 2005)

    Google Scholar 

  21. R.S. Strichartz, The Campbell–Baker–Hausdorff–Dynkin formula and solutions of differential equations. J. Funct. Anal. 72, 320–345 (1987)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The research on this paper was partially supported by the Norwegian Research Council (project 231632).

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Authors and Affiliations

  1. Norwegian University of Science and Technology (NTNU), Institutt for matematiske fag, Trondheim, Norway

    Charles Curry & Kurusch Ebrahimi-Fard

  2. Univ. Côte d’Azur, CNRS, UMR 7351, Nice Cedex, France

    Frédéric Patras

Authors
  1. Charles Curry
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  2. Kurusch Ebrahimi-Fard
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  3. Frédéric Patras
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Corresponding author

Correspondence to Kurusch Ebrahimi-Fard .

Editor information

Editors and Affiliations

  1. University of Bergen, Bergen, Norway

    Florin Adrian Radu

  2. University of Bergen, Bergen, Norway

    Kundan Kumar

  3. University of Bergen, Bergen, Norway

    Inga Berre

  4. University of Bergen, Bergen, Norway

    Jan Martin Nordbotten

  5. Hasselt University, Diepenbeek, Belgium

    Iuliu Sorin Pop

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Curry, C., Ebrahimi-Fard, K., Patras, F. (2019). On Non-commutative Stochastic Exponentials. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_39

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