  We derive the generalized Fokker-Planck equation associated with the Langevin
equation (in the Ito sense) for an overdamped particle in an external potential
driven by multiplicative noise with an arbitrary distribution of the increments
of the noise generating process. We explicitly consider this equation for
various specific types of noises, including Poisson white noise and L\'{e}vy
stable noise, and show that it reproduces all Fokker-Planck equations that are
known for these noises. Exact analytical, time-dependent and stationary
solutions of the generalized Fokker-Planck equation are derived and analyzed in
detail for the cases of a linear, a quadratic, and a tailored potential.
