The third order superharmonic response of a Duffing oscillator to narrow band random excitation is analyzed. The analysis shows the effect of excitation bandwidth on the response and stability of the non-linear oscillator. In particular the analysis shows that multivalued superharmonic response can occur only when the excitation bandwidth is small. Stability of the response is examined by constructing a mean square phase plane from a time dependent Fokker-Planck equation and also by perturbing the stationary solutions.