Weighted norm inequalities and hedging in incomplete markets

Abstract

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 Holder inequalities which are equivalent to weighted norm inequalities. By means of these last inequalities, we also extend previous results on the Follmer-Schweizer decomposition.