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4 matches found
XIII: 26, 294-306, LNM 721 (1979)
BONAMI, Aline; LÉPINGLE, Dominique
Fonction maximale et variation quadratique des martingales en présence d'un poids (Martingale theory)
Weighted norm inequalities in martingale theory assert that a martingale inequality---relating under the law $P$ two functionals of a $P$-martingale---remains true, possibly with new constants, when $P$ is replaced by an equivalent law $Z.P$. To this order, the weight'' $Z$ must satisfy special conditions, among which a probabilistic version of Muckenhoupt's (1972) $(A_p)$ condition and a condition of multiplicative boundedness on the jumps of the martingale $E[Z\,|\,{\cal F}_t]$. This volume contains three papers on weighted norms inequalities, 1326, 1327, 1328, with considerable overlap. Here the main topic is the weighted-norm extension of the Burkholder-Gundy inequalities
Comment: Recently (1997) weighted norm inequalities have proved useful in mathematical finance
Keywords: Weighted norm inequalities, Burkholder inequalities
Nature: Original
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XIII: 27, 307-312, LNM 721 (1979)
IZUMISAWA, Masataka; SEKIGUCHI, Takesi
Weighted norm inequalities for martingales (Martingale theory)
See the review of 1326. The topic is the same, though the proof is different
Comment: See the paper by Kazamaki-Izumisawa in Tôhoku Math. J. 29, 1977. For a modern reference see also Kazamaki, Continuous Exponential Martingales and $\,BMO$, LNM. 1579, 1994
Keywords: Weighted norm inequalities, Burkholder inequalities
Nature: Original
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XIII: 28, 313-331, LNM 721 (1979)
Comment: An exponent $1/\lambda$ is missing in formula (4), p.315
Chevalier has strengthened the Burkholder inequalities into an equivalence of $L^p$ norms between $M^{\ast}\lor Q(M)$ and $M^{\ast}\land Q(M)$, where $M$ is a martingale, $M^{\ast}$ is its maximal function and $Q(M)$ its quadratic variation. This has been extended to all moderate Orlicz spaces in 1404. The present paper further extends the result to the Orlicz spaces of a law $\widehat P$ equivalent to $P$, provided the density is an $(A_p)$ weight (see 1326)