We determine the variance-optimal hedge when the logarithm of the underlying price follows a process with stationary independent increments in discrete or continuous time. Although the general solution to this problem is known as backward recursion or backward stochastic differential equation, we show that for this class of processes the optimal endowment and strategy can be expressed more explicitly. The corresponding formulas involve the moment, respectively, cumulant generating function of the underlying process and a Laplace-or Fourier-type representation of the contingent claim. An example illustrates that our formulas are fast and easy to evaluate numerically.

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L'accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Closedness of some spaces of stochastic integrals

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 BurkholderDavis-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 1 d -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.Mathematics Subject classification (1991): 60G48, 60H05, 90A09