  Hybrid Inflation is a two-field model where inflation ends by a tachyonic
instability, the duration of which is determined by stochastic effects and has
important observational implications. Making use of the recursive approach to
the stochastic formalism presented in Ref. [1], these effects are consistently
computed. Through an analysis of back-reaction, this method is shown to
converge in the valley but points toward an (expected) instability in the
waterfall. It is further shown that quasi-stationarity of the auxiliary field
distribution breaks down in the case of a short-lived waterfall. It is found
that the typical dispersion of the waterfall field at the critical point is
then diminished, thus increasing the duration of the waterfall phase and
jeopardizing the possibility of a short transition. Finally, it is found that
stochastic effects worsen the blue tilt of the curvature perturbations by an
order one factor when compared with the usual slow-roll contribution.
