We have analyzed the phenomenon of stochastic resonance in an asymmetric bistable system driven by multiplicative non-Gaussian noise and additive Gaussian noise. Using a path-integral approach, together with the unified colored noise approximation and two-state model theory, we have obtained a consistent Markovian approximation, which enables us to get the analytical expressions for the stationary probability distribution and the signal-to-noise ratio. Under the influence of non-Gaussian noise deviation parameter, noise correlation time, asymmetric coefficient and mutual correlation strength, there are stochastic resonance on signal-to-noise ratio as non-Gaussian noise intensity and Gaussian noise intensity. Besides, the influence of different parameters on signal-to-noise ratio is discussed respectively, including non-Gaussian noise deviation parameter, correlation times of the non-Gaussian noise, cross-correlation strengths, amplitudes of periodic signal, and asymmetric coefficient.
