Weighted Norm Inequalities and Closedness of a Space of Stochastic Integrals

Let X be an IR-valued special semimartingale on a probability space (Ω,F , (Ft)0≤t≤T , P ) with canonical decomposition X = X0 + M + A. Denote by GT (Θ) the space of all random variables (θ ·X)T , where θ is a predictable X-integrable process such that the stochastic integral θ ·X is in the space S2 of semimartingales. We investigate under which conditions on the semimartingale X the space GT (Θ) is closed in L2(Ω,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 GT (Θ) in L2(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.

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