APPROXIMATING RANDOM-VARIABLES BY STOCHASTIC INTEGRALS

Let X be a semimartingale and Theta the space of all predictable X-int egrable processes theta such that integral theta dX is in the space S- 2 of semimartingales, we consider the problem of approximating a given random variable H epsilon L(2) by a stochastic integral integral(0)(T ) theta(s) dX(s), with respect to the L(2)-norm. If X is special and h as the form X = X(0) +M+ integral alpha d(M), we construct a solution in feedback form under the assumptions that integral alpha(2)d(M) is d eterministic and that H admits a strong F-S decomposition Into a const ant, a stochastic integral of X and a martingale part orthogonal to M. We provide sufficient conditions for the existence of such a decompos ition, and we give several applications to quadratic optimization prob lems arising in financial mathematics.