On L²-projections on a space of stochastic integrals

Let X be an Rᵈ-valued continuous semimartingale, T a fixed time horizon and Θ the space of all Rᵈ -valued predictable X -integrable processes such that the stochastic integral G(ϑ)=∫ϑ dX is a square-integrable semimartingale. A recent paper gives necessary and sufficient conditions on X for GT(Θ) to be closed in L²(P). In this paper, we describe the structure of the L²-projection mapping an FT-measurable random variable H ∈ L²(P) on GT(θ) and provide the resulting integrand ϑH ∈ Θ 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 P for X and weighted norm inequalities relating P to the original measure P.