Variance-optimal hedging in discrete-time

We solve the problem of approximating in L2 a given random variable H by stochastic integrals G(T)(I) of a given discrete-time process X. We interpret H as a contingent claim to be paid out at time T, X as the price evolution of some risky asset in a financial market and G(I) as the cumulative gains from trade using the hedging strategy I. As an application we determine the variance-optimal strategy which minimizes the variance of the net loss H – G(T)(I) over all strategies I.