Weighted norm inequalities and hedging in incomplete markets

Let $X$ be an ${\Bbb R}^d$-valued special semimartingale on a probability space $(\Omega , {\cal F} , ({\cal F} _t)_{0 \leq t \leq T} ,P)$ with canonical decomposition $X=X_0+M+A$. Denote by $G_T(\Theta )$ the space of all random variables $(\theta \cdot X)_T$, where $\theta $ is a predictable $X$-integrable process such that the stochastic integral $\theta \cdot X$ is in the space ${\cal S} ^2$ of semimartingales. We investigate under which conditions on the semimartingale $X$ the space $G_T(\Theta )$ is closed in ${\cal L} ^2(\Omega , {\cal F} ,P)$, a question which arises naturally in the applications to financial mathematics. Our main results give necessary and/or sufficient conditions for the closedness of $G_T(\Theta )$ in ${\cal L} ^2(P)$. Most of these conditions deal with BMO-martingales and reverse Hölder inequalities which are equivalent to weighted norm inequalities. By means of these last inequalities, we also extend previous results on the Föllmer-Schweizer decomposition.