## 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

### Keywords:

Burkholder-Davis-Gundy inequalities; local martingale
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