  We present a detailed study of the statistical properties of an Agent Based
Model and of its generalization to the multiplicative dynamics. The aim of the
model is to consider the minimal elements for the understanding of the origin
of the Stylized Facts and their Self-Organization. The key elements are
fundamentalist agents, chartist agents, herding dynamics and price behavior.
The first two elements correspond to the competition between stability and
instability tendencies in the market. The herding behavior governs the
possibility of the agents to change strategy and it is a crucial element of
this class of models. The linear approximation permits a simple interpretation
of the model dynamics and, for many properties, it is possible to derive
analytical results. The generalized non linear dynamics results to be extremely
more sensible to the parameter space and much more difficult to analyze and
control. The main results for the nature and Self-Organization of the Stylized
Facts are, however, very similar in the two cases. The main peculiarity of the
non linear dynamics is an enhancement of the fluctuations and a more marked
evidence of the Stylized Facts. We will also discuss some modifications of the
model to introduce more realistic elements with respect to the real markets.
