Quantile Optimization Under Derivative Constraint

Abstract

This paper studies a new type of quantile optimization problems arising from insurance contract design models. This type of optimization problems is characterized by a constraint of infinity-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 hazard in 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.