Minimizing expected loss of hedging in incomplete and constrained markets

We study the problem of minimizing the expected discounted loss E[e(-)integral(o)(T) (r(u)du) (C - X-x,X-pi(T))(+)] when hedging a liability C at time t = T, using an admissible portfolio str ategy pi(.) and starting with initial wealth x. The existence of an optimal solution is established in the context of continuous-time Ito process inco mplete market models, by studying an appropriate dual problem. It is shown that the optimal strategy is of the form of a knock-out option with payoff C, where the domain of the knock-out depends on the value of the optimal du al variable. We also discuss a dynamic measure for the risk associated with the liability C, defined as the supremum over different scenarios of the m inimal expected loss of hedging C.