Weighted Norm Inequalities and Hedging in Incomplete Markets

Let X be an R^d-valued special semimartingale on a probability space with canonical decomposition X=X_0+M+A. Denote by G_T(Theta) the space of all random variables (theta bullet X)_T, where theta is a predictable X- integrable process such that the stochastic integral theta bullet X is in the space S^2 of semimartingales. We investigate under which conditions on the semimartingale X the space G_T(Theta) is closed in L^2(Omega,F,P), a question which arises naturally in the applications to financial mathematics. Our main results give necessary and/or sufficient conditions for the closedness of G_T(Theta) inL^2(P). Most of these conditions deal with BMO-martingales and reverse Hoelder inequalities which are equivalent to weighted norm inequalities. By means of these last inequalities, we also extend previous results on the Foellmer-Schweizer decomposition