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Sur certaines généralisations de l’inégalité de Fefferman. (French) Zbl 0542.60047

Sémin. probabilités XVIII, 1982/83, Proc., Lect. Notes Math. 1059, 219-222 (1984).
[For the entire collection see Zbl 0527.00020.]
An extension of Fefferman’s inequality is given: suppose that X,Y are two semimartingales, the random variable \([Y,Y]_{\infty}\) is finite a.s., then \[ E\int^{\infty}_{0}| d[X,Y]_ s| \leq c E(\int^{\infty}_{0}Z^*_ sd[X,X]_ s)^{1/2} \] where \(Z^*_ t=\sup_{s\leq t}| Z_ s|\), Z is a unique optional positive process such that \(Z^ q_ T=E[([Y,Y]_{\infty}-[Y,Y]_{T-})^ q| F_ T]\) for any stopping time T and some \(q>0\). The constant c depends only on q. An extension of the inequality is given for an increasing moderate function and for a predictable case in which the process [\(\cdot,\cdot]\) is replaced by \(<\cdot,\cdot>\) when the last process exists.
Reviewer: L.I.Gal’chuk

MSC:

60G44 Martingales with continuous parameter
60H05 Stochastic integrals

Citations:

Zbl 0527.00020
Full Text: Numdam EuDML