Follmer-Schweizer Decomposition and Mean-Variance Hedging for General Claims

Let X be an Rᵈ-valued special semimartingale on a probability space (Ω, F, (Fₜ)0≤ t ≤ T,P) with decomposition X = X₀ + M + A and Θ the space of all predictable, X-integrable processes θ such that ∫θ dX is in the space J² of semimartingales. If H is a random variable in L², we prove, under additional assumptions on the process X, that H can be written as the sum of an F₀-measurable random variable H₀, 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 L²-norm. Finally, we deduce from this continuity that the subspace of L² generated by ∫θ dX, where θ∈ Θ, is closed in L², and we give some applications of this result to financial mathematics.