Before revision:  This type of optimization problems is characterized by a constraint ofinfinity -dimension, that is, the derivatives of the decision quantile functions are bounded. Such a constraint essentially comes from the "incentive compatibility" constraint for any optimal insurance contract to avoid the potential severe problem of moral hazardin insurance contract design models. By a further development of the author's relaxation method, this paper provides a systemic approach to solving this new type of quantile optimization problems }  .  The optimal quantile is expressed via the solution of a free boundary problem for a  second-order nonlinear ordinary differential equation (ODE), which is similar to the Black-Scholes ODE for perpetual American options and has been well studied in literature theoretically and numerically .
After revision:  This type of optimization problems is characterized by a constraint ofinfinity problem under rank-dependent utility (RDU) theory. Their results, however, suffer from a moral hazard problem, namely, providing incentives for the insured to falsely report the actual loss. Xu et al. (2019) investigated the same problem, but took the } . The optimal quantile is expressed via the solution of a free boundary problem for a second-order nonlinear ordinary differential equation (ODE), which is similar to the Black-Scholes ODE for perpetual American options and has been well studied in literature theoretically and numerically .