  We analyze the properties of arguably the simplest bilinear stochastic
multiplicative process, proposed as a model of financial returns and of other
complex systems combining both nonlinearity and multiplicative noise. By
construction, it has no linear predictability (zero two-point correlation) but
a certain nonlinear predictability (non-zero three-point correlation). It can
thus be considered as a paradigm for testing the existence of a possible
nonlinear predictbility in a given time series. We present a rather exhaustive
study of the process, including its ability to produce fat-tailed distribution
from Gaussian innovations, the unstable characteristics of the inversion of the
key nonlinear parameters and of the two initial conditions necessary for the
implementation of a prediction scheme and an analysis of the associated
super-exponential sensitivity of the inversion of the innovations in the
presence of a large impluse. Our study emphasizes the conditions under which a
degree of predictability can be achieved and describes a number of different
attempts, which overall illuminates the properties of the process. In
conclusion, notwithstanding its remarkable simplicity, the bilinear stochastic
process exhibits remarkably rich and complex behavior, which makes it a serious
candidate for the modeling of financial times series and of other complex
systems.
