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Remarks on the Burkholder-Davis-Gundy inequalities. (Remarques sur les inégalités de Burkholder-Davis-Gundy.) (French) Zbl 0821.60053

Azéma, Jacques (ed.) et al., Séminaire de Probabilités XXVIII. Berlin: Springer-Verlag. Lect. Notes Math. 1583, 92-97 (1994).
If \(p \geq 1\), Burkholder-Davis-Gundy inequalities tell us there exist \(0 < c_ p < C_ p < \infty\) such that for any local martingale \(M\), \[ (1) \quad c_ p \| M^*_ \infty \|_ p \leq \bigl \| [M,M]^{1/2}_ \infty \bigr \|_ p; \qquad (2) \quad \bigl \| [M,M]^{1/2}_ \infty \bigr \|_ p \leq C_ p \| M^*_ \infty \|_ p, \] where \(M^*_ \infty = \sup_{t \geq 0} | M_ t |\). Moreover if \(M\) is continuous, (1) and (2) hold for any \(p > 0\). If \(0 < p < 1\), the author gives two explicit and very simple (no- continuous) martingales \(M\) such that (1) and (2) are not realized.
For the entire collection see [Zbl 0797.00020].
Reviewer: P.Vallois (Nancy)

MSC:

60G44 Martingales with continuous parameter
60E15 Inequalities; stochastic orderings
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