Let (M, g) be a Lorentzian warped product space-timeM=(a, b)×H, g = −dt2 ⊕fh, where −∞⩽a −∞ and (H, h) is homogeneous, then the past incompleteness of every timelike geodesic of (M,g) is stable under smallC0 perturbations in the space Lor(M) of Lorentzian metrics forM. Also we show that if (H,h) is isotropic and (M,g) contains a past-inextendible, past-incomplete null geodesic, then the past incompleteness of all null geodesics is stable under smallC1 perturbations in Lor(M). Given either the isotropy or homogeneity of the Riemannian factor, the background space-time (M,g) is globally hyperbolic. The results of this paper, in particular, answer a question raised by D. Lerner for big bang Robertson-Walker cosmological models affirmatively.