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The set-indexed Ito integral

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Abstract

We construct an Ito-type stochastic integral where the integrator is a process indexed by a semilattice of compact subsets of a fixed topological spaceT and the integrands, which are indexed by the points inT, possess a natural form of predictability. The definition of the integral involves, among other things, an Ito-type isometry defined in terms of the set-indexed quadratic variation of the integrator. The martingale property and quadratic variation for the resulting integral process are derived. In addition, employing the notion of stopping set from Ivanoff and Merzbach (1995), we construct and study a set-indexed local integral. A novel and flexible notion of predictability for set-indexed processes is defined and characterized, permitting the integration of a set-indexed integrand against a set-indexed process.

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Authors and Affiliations

  1. Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, 91905, Jerusalem, Israel

    Diane Saada

  2. Department of Statistics, University of Manitoba, R3T 2N2, Winnipeg, Manitoba, Canada

    Dean Slonowsky

Authors
  1. Diane Saada
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  2. Dean Slonowsky
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Corresponding author

Correspondence to Diane Saada.

Additional information

Research supported in part by the Israel Science Foundation (grant no.: 0321423).

Research supported in part by a grant from the Natural, Sciences and Engineering Research Council of Canada.

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Saada, D., Slonowsky, D. The set-indexed Ito integral. J. Anal. Math. 94, 61–89 (2004). https://doi.org/10.1007/BF02789042

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  • DOI: https://doi.org/10.1007/BF02789042

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