Abstract
In a previous paper we introduced a new concept, the notion of ℰ-martingales and we extended the well-known Doob inequality (for 1 < p < + ∞) and the Burkholder–Davis–Gundy inequalities (for p = 2) to ℰ-martingales. After showing new Fefferman-type inequalities that involve sharp brackets as well as the space bmo q , we extend the Burkholder–Davis–Gundy inequalities (for 1 < p < + ∞) to ℰ-martingales. By means of these inequalities we give sufficient conditions for the closedness in L p of a space of stochastic integrals with respect to a fixed ℝd-valued semimartingale, a question which arises naturally in the applications to financial mathematics. Finally we investigate the relation between uniform convergence in probability and semimartingale topology.
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Received: 22 July 1997 / Revised version: 3 July 1998
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Choulli, T., Stricker, C. & Krawczyk, L. On Fefferman and Burkholder–Davis–Gundy inequalities for ℰ-martingales. Probab Theory Relat Fields 113, 571–597 (1999). https://doi.org/10.1007/s004400050218
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DOI: https://doi.org/10.1007/s004400050218
- Mathematics Subject classification (1991): 60G48, 60H05, 90A09
- Keywords: Semimartingales, Semimartingale topology, Stochastic integrals, Stochastic Exponential, Fefferman inequality, Burkholder–Davis–Gundy inequality.