We compute the expectation and the two-point correlation of the solution to elliptic boundary value problems with stochastic input data. Besides stochastic loadings, via perturbation theory, our approach covers also elliptic problems on stochastic domains or with stochastic coefficients. The solution's two-point correlation satisfies a deterministic boundary value problem with the two-fold tensor product operator on the two-fold tensor tensor product domain. We discuss the efficient solution of such tensor product problems by either a sparse grid approach based on multilevel frames or by the pivoted Cholesky decomposition. Both approaches involve only standard finite element techniques. Numerical results illustrate the algorithms.