Mutual optical bistability is found for several modes in open cavities containing second- or third-order nonlinear materials. The quantum-statistical treatment yields the Fokker-Planck equation for any system of damped boson modes. Under certain conditions it is possible to state the extrema of the steady solution of the Fokker-Planck equation. They are compared with the steady states resulting from the classical deterministic treatment. In some cases both statements coincide.