The Cheapest Superstrategy without Optional Decomposition

We follow very closely the Föllmer and Kabanov Lagrange multiplier approach to superstrategies in perfect incomplete markets, except that we provide a very simple proof of the existence of a minimizing multiplier in the case of a European option under the assumption that the discounted process of the underlying is an $L^{2}(P)$-martingale for some probability $P$. Even if it gives the existence of a superstrategy associated with the supremum of the expectations under equivalent martingale measures, our result is much weaker than the optional decomposition theorem.