A kind of shape memory alloy (SMA) hysteretic nonlinear model is developed, and the stochastic bifurcation characteristics of SMA intravascular stents subjected to radial and axial excitations are studied in this paper. A new nonlinear differential item is introduced to interpret the hysteretic phenomena of SMA strain-stress curves, and the dynamic model of SMA intravascular stent subjected to radial and axial stochastic excitations is established. The conditions of the system's stochastic stability are determined, and the probability density function of the system response is obtained. Finally, the stochastic Hopf bifurcation characteristics of the system are analyzed. Theoretical analysis and numerical simulation show that the system stability varies with bifurcation parameters, and stochastic Hopf bifurcation occurs in the process; there are two limit cycles in the stationary probability density of the system response in some cases, which means that there are two vibration amplitudes whose probability are both very high; jumping phenomena between the two vibration amplitudes appears with the change of conditions, which may cause stent fracture or loss. The results of this paper are helpful for application of SMA intravascular stent in biomedical engineering fields.