Computing the expected value of a concave, piecewise affine bivariate function for the gamma distribution

We present a method for exact computation of the expected value of a non-separable, piecewise linear function of two variables. We assume a stochastic model of two random variables obtained by linear combination of two independent principal components having gamma distributions with integer shape parameters. The method can be implemented so the computational effort is proportional to the number of piecewise affine pieces times the product of the shape parameters of the gamma distributions. It can be used for fast computation of expected values in stochastic dynamic programming.