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 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 .
Bernard et al. (2015) studied an insurance contract design 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 incentive compatibility} constraint into consideration to avoid that moral hazard . Mathematically speaking, the model reduces to a quantile optimisation problem with a compatibility constraint. They solved the problem by imposing assumptions on the loss and the probability weighting function. This paper solves the problem completely by a new quantile optimisation approach under general setting. The optimal solution is expressed by the solution of an obstacle problem for a semilinear second-order elliptic operator with mixed boundary conditions. Surprisingly, it is shown that every reasonable moral-hazard-free contract is optimal for infinitely many RDU maximisers with different utility functions and probability weighting functions .
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This paper studies a new type of quantile optimization problems arising from
Bernard et al. (2015) studied an
D
models. This type of optimization problems is characterized by a constraint of
D
infinity
R
-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
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
A
incentive compatibility
A
constraint into consideration to avoid that moral hazard
A
Mathematically speaking, the model reduces to a quantile optimisation problem with a compatibility constraint. They solved the problem by imposing assumptions on the loss and the probability weighting function. This paper solves the problem completely by a new quantile optimisation approach under general setting.
R
quantile is expressed via
solution is expressed by
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a free boundary
an obstacle
A
semilinear
R
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
elliptic operator with mixed boundary conditions. Surprisingly, it is shown that every reasonable moral-hazard-free contract is optimal for infinitely many RDU maximisers with different utility functions and probability weighting functions