Convex Duality in Mean-Variance Hedging Under Convex Trading Constraints

We study mean-variance hedging under portfolio constraints in a general semi-martingale model. The constraints are formulated via predictable correspondences,meaning that the trading strategy is restricted to lie in a closed convex set whichmay depend on the state and time in a predictable way. To obtain the existence ofa solution, we first establish the closedness in L2 of the space of all gains from trade(i.e., the terminal values of stochastic integrals with respect to the price process ofthe underlying assets). This is a first main contribution which enables us to tacklethe problem in a systematic and unified way. In addition, using the closedness al-lows us to explain and generalise in a systematic way the convex duality resultsobtained previously by other authors via ad hoc methods in specific frameworks.[...]