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Strong Martingales: Their Decompositions and Quadratic Variation

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  • Dean Slonowsky

    (University of Manitoba)

Abstract

Set-indexed strong martingales and a form of predictability for set-indexed processes are defined. Under a natural integrability condition, we show that any set-indexed strong submartingale can be decomposed in the Doob–Meyer sense. A form of predictable quadratic variation for square-integrable set-indexed strong martingales is defined and sufficient conditions for its existence are given. Under a conditional independence assumption, these reduce to a simple moment condition and, if the strong martingale has continuous sample paths, the resulting quadratic variation can be approximated in the L 2-sense by sums of conditional expectations of squared increments.

Suggested Citation

  • Dean Slonowsky, 2001. "Strong Martingales: Their Decompositions and Quadratic Variation," Journal of Theoretical Probability, Springer, vol. 14(3), pages 609-638, July.
    • Handle: RePEc:spr:jotpro:v:14:y:2001:i:3:d:10.1023_a:1017536921656
    DOI: 10.1023/A:1017536921656
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    References listed on IDEAS

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    1. Ivanoff, B. Gail & Merzbach, Ely, 1995. "Stopping and set-indexed local martingales," Stochastic Processes and their Applications, Elsevier, vol. 57(1), pages 83-98, May.
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