  We present the first detailed analysis of the statistical properties of jump
processes bounded by a saturation function and driven by Poisson white noise,
being a random sequence of delta pulses. The Kolmogorov-Feller equation for the
probability density function (PDF) of such processes is derived and its
stationary solutions are found analytically in the case of the symmetric
uniform distribution of pulse sizes. Surprisingly, these solutions can exhibit
very complex behavior arising from both the boundedness of pulses and
processes. We show that all features of the stationary PDF (number of branches,
their form, extreme values probability, etc.) are completely determined by the
ratio of the saturation function width to the half-width of the pulse-size
distribution. We verify all theoretical results by direct numerical
simulations.
