On the shortfall risk control -- a refinement of the quantile hedging method

The issue of constructing a risk minimizing hedge with additional constraints on the shortfall risk is examined. Several classical risk minimizing problems have been adapted to the new setting and solved. The existence and specific forms of optimal strategies in a general semimartingale market model with the use of conditional statistical tests have been proven. The quantile hedging method applied in \cite{FL1} and \cite{FL2} as well as the classical Neyman-Pearson lemma have been generalized. Optimal hedging strategies with shortfall constraints in the Black-Scholes and exponential Poisson model have been explicitly determined.