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Discrete stochastic integration in Riesz spaces

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Abstract

In this work we continue the developments of Kuo et al. (Indag Math 15:435–451, 2004; J Math Anal Appl 303:509–521, 2005) with the construction of the martingale transform or discrete stochastic integral in a Riesz space (measure-free) setting. The discrete stochastic integral is considered both in terms of a weighted sum of differences and via bilinear vector-valued forms. For this, analogues of the spaces L 2 and Mart2 on Riesz spaces with a conditional expectation operator and a weak order unit are constructed using the f-algebra structure of the universal completion of the Riesz space and properties of the extension of the conditional expectation to its natural domain.

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Authors and Affiliations

  1. School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, PO, Wits, 2050, South Africa

    Coenraad C. A. Labuschagne

  2. School of Mathematics, University of the Witwatersrand, Private Bag 3, PO, Wits, 2050, South Africa

    Bruce A. Watson

Authors
  1. Coenraad C. A. Labuschagne
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  2. Bruce A. Watson
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Correspondence to Bruce A. Watson.

Additional information

C. C. A. Labuschagne was supported in part by the South African National Research Foundation grant FA2007041200015 and B. A. Watson was supported in part by the John Knopfmacher Centre for Applicable Analysis and Number Theory and by the South African National Research Foundation grant FA2007041200006.

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Labuschagne, C.C.A., Watson, B.A. Discrete stochastic integration in Riesz spaces. Positivity 14, 859–875 (2010). https://doi.org/10.1007/s11117-010-0089-1

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