We define the higher order moments associated to the stochastic solution of an elliptic BVP in D^d with stochastic input data. We prove that the k-th moment solves a deterministic problem in D^kdk, for which we discuss well-posedness and regularity. We discretize the deterministic k-th moment problem using sparse grids and, exploiting a spline wavelet basis, we propose an efficient algorithm, of logarithmic-linear complexity, for solving the resulting system.