Shortfall risk minimization under model uncertainty in the binomial case: adaptive and robust approaches

We consider the problem of minimizing the shortfall risk when the aim is to hedge a contingent claim in a binomial market model and the initial capital is insufficient for a perfect hedge. This problem has been solved under complete information on the underlying model in [3]. We present two possible solutions to the same problem in the case of incomplete information, namely when the underlying probability measure is unknown. The results obtained can also be applied to other classical problems, such as VaR minimization or maximum loss minimization. Key-Words: shortfall risk minimization, binomial model, adaptive control, robust control.