FOLLMER-SCHWEIZER DECOMPOSITION AND MEAN-VARIANCE HEDGING FOR GENERAL CLAIMS

Let X be an R(d)-valued special semimartingale on a probability space (Omega,F,(F-t)(0 less than or equal to t less than or equal to T), P) with decomposition X = X(0) + M + A and Theta the space of all predictable, X-integrable processes theta such that integral theta dX is in the space P-2 of semimartingales. If H is a random variable in L(2), We prove, under additional assumptions on the process X, that H can be written as the sum of an F-0-measurable random variable H-0, a stochastic integral of X and a martingale part orthogonal to M. Moreover, this decomposition is unique and the function mapping H with its decomposition is continuous with respect to the L(2)-norm. Finally, we deduce from this continuity that the subspace of L(2) generated by integral theta dX, where theta epsilon Theta, is closed in L(2), and we give some applications of this result to financial mathematics.