Smooth approximation of the quantile function derivatives

In this paper, a smooth approximation of the second-order derivatives of quantile function is provided. The convergence of approximations of the first and second order derivatives of quantile function is studied in cases when there exists a deterministic equivalent for the corresponding stochastic programming problem. The quantile function is one of common criteria in stochastic programming problems. The first-order derivative of quantile function can be represented as a ratio of partial derivatives of probability function. Using smooth approximation of probability function and its derivatives we obtain approximations of these derivatives in the form of volume integrals. Approximation of the second-order derivative is obtained directly as derivative of the first-order derivative. A numerical example is provided to evaluate the accuracy of the presented approximations.