On L2-projections on a space of stochastic integrals

Let X be an IR-valued continuous semimartingale, T a fixed time horizon and Θ the space of all IR-valued predictable X-integrable processes such that the stochastic integral G(θ) = ∫ θdX is a square-integrable semimartingale. A recent paper of Delbaen/Monat/Schachermayer/Schweizer/Stricker (1996) gives necessary and sufficient conditions on X for GT (Θ) to be closed in L(P ). In this paper, we describe the structure of the L-projection mapping an FT measurable random variable H ∈ L(P ) on GT (Θ) and provide the resulting integrand θ ∈ Θ in feedback form. This is related to variance-optimal hedging strategies in financial mathematics and generalizes previous results imposing very restrictive assumptions on X. Our proofs use the variance-optimal martingale measure P̃ for X and weighted norm inequalities relating P̃ to the original measure P .