Before revision: In that paper, we solve dynamically a partial hedging problem for an American contingent claim: assuming superhedging is not feasible, we explain in this context the notion of efficient hedging by introducing a risk minimization criterion: we consider here the problem of minimizing the conditional expected loss for a given convex and non decreasing loss function. To solve this problem, we provide a connection between the dynamic convex risk functional introduced and the solutionof a quadratic RBSDE (Reflected Backward Stochastic Differential Equations): this is achieved by studying the properties of specific non linear expectations  .
After revision: In that paper, we provide a connection between the dynamic convex risk functional introduced and the solutionof a quadratic RBSDE (Reflected Backward Stochastic Differential Equations): this is achieved by studying the properties of specific non linear expectations .