The Landau-Lifschitz fluctuating hydrodynamics formalism is applied to study the statistical properties of simple fluids in a finite system under nonequilibrium constraints. The boundary conditions are explicitly taken into account so that the results can be compared with particle simulations. Two scenarios are investigated: a fluid subjected to a constant shear and a fluid subjected to a constant temperature gradient. By considering a fluid with vanishing thermal expansivity, exact results are obtained for the static and dynamic correlation functions.