ON L-2 PROJECTIONS ON A SPACE OF STOCHASTIC INTEGRALS

Let X be an R-d-valued continuous semimartingale, T a fixed time horiz on and Theta the space of all R-d-valued predictable X-integrable proc esses such that the stochastic integral G(theta) = integral theta dX i s a square-integrable semimartingale. A recent paper gives necessary a nd sufficient conditions on X for G(T)(Theta) to be closed in L-2(P). In this paper, we describe the structure of the L-2-projection mapping an F-T-measurable random variable H is an element of L-2(P) on GT(The ta) and provide the resulting integrand theta(H) is an element of Thet a in feedback form. This is related to variance-optimal hedging strate gies in financial mathematics and generalizes previous results imposin g very restrictive assumptions on X. Our proofs use the variance-optim al martingale measure (Pq) over tilde for X and weighted norm inequali ties relating (P) over tilde to the original measure P.