Аннотация:The following study investigates the problems of portfolio optimization under partial hedging of contingent claims. We consider arbitrage-free incomplete discrete markets representing them in form of scenario trees. Two well-known problems of quantile hedging and hedging with minimal risk of shortfall are analyzed. We discuss methods of solving them for two cases of European and American contingent claims applying some decomposition techniques and principles of optimality. Besides, the dynamic programming algorithm is described to build the superhedging strategy.