Stochastic programming duality: 8 multipliers for unbounded constraints with an application to mathematical finance

A new duality theory is developed for a class of stochastic programs in which the probability distribution is not necessarily discrete. This provides a new framework for problems which are not necessarily bounded, are not required to have relatively complete recourse, and do not satisfy the typical Slater condition of strict feasibility. These problems instead satisfy a different constraint qualification called ‘direction-free feasibility’ to deal with possibly unbounded constraint sets, and ‘calmness’ of a certain finite-dimensional value function to serve as a weaker condition than strict feasibility to obtain the existence of dual multipliers. In this way, strong duality results are established in which the dual variables are finite-dimensional, despite the possible infinitedimensional character of the second-stage constraints. From this, infinite-dimensional dual problems are obtained in the space of essentially bounded functions. It is then shown how this framework could be used to obtain duality results in the setting of mathematical finance.