Yang, X.

state problem portfolio constraint solution state price density price condition var constraint example investor probability state splitting problem 1 security innovation allocation splitting utility density order insurance portfolio allocation unconstrained partition problem budget constraint level state space market figure order condition portfolio insurance var level risk constraint value function number portfolio insurance problem nphard monotonic polynomial budget insurance constraint problem 1. state i solution x journal monotonicity paper partition p 2 /2 p 1 /1 objective function value claim downside risk constraint space portfolio problem split proof unconstrained case consumption lottery respect downside state 1 risk constraints uniform section polynomial time theorem restriction utility function incentive 1/100 1/2 choice swedish lottery bonds polynomial time solution figure 1. i =1 i