On L2-projections on a space of stochastic integrals

Abstract

Let X be an ℝd-valued continuous semimartingale, T a fixed time horizon and Θ the space of all ℝd-evalued 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 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.