Abstract: The possibility for onset of chaotic motion in the Duffing oscillator under parametric excitation of bounded noise is studied The stochastic Melnikov process is first derived and the critical value of excitation amplitude for the onset of chaotic motion is obtained based on the stochastic Melnikov process having simple zero in the mean square sense. Then,the largest Lyapunov exponent of the system is calculated numerically and the critical value of excitation amplitude is obtained based on vanishing of the largest Lyapunov exponent. It is found that both two critical values increase as the intensity of noise increases for larger value of noise intensity.