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July, 1987 Quadratic Variation for a Class of $L \log^+ L$-Bounded Two-parameter Martingales
Nikos E. Frangos, Peter Imkeller
Ann. Probab. 15(3): 1097-1111 (July, 1987). DOI: 10.1214/aop/1176992083


Let $M = (M_t)_{t\in\lbrack 0,1\rbrack^2}$ be a two-parameter $L\log^+ L$-bounded (not necessarily continuous) martingale. Assume that the marginal filtrations in the first and second directions are quasi-left continuous. We prove the existence of quadratic variation in the sense of convergence in probability. This is done first for bounded martingales. The extension to the general case is obtained by approximating a given martingale by its bounded truncations and using a two-parameter version of the square function inequality of Burkholder.


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Nikos E. Frangos. Peter Imkeller. "Quadratic Variation for a Class of $L \log^+ L$-Bounded Two-parameter Martingales." Ann. Probab. 15 (3) 1097 - 1111, July, 1987.


Published: July, 1987
First available in Project Euclid: 19 April 2007

zbMATH: 0631.60048
MathSciNet: MR893916
Digital Object Identifier: 10.1214/aop/1176992083

Primary: 60G44
Secondary: 60G07 , 60G42

Keywords: Burkholder inequality , dual previsible projections , Quadratic Variation , Two-parameter martingales

Rights: Copyright © 1987 Institute of Mathematical Statistics

Vol.15 • No. 3 • July, 1987
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