  In science and especially in economics, agent-based modeling has become a
widely used modeling approach. These models are often formulated as a large
system of difference equations. In this study, we discuss two aspects,
numerical modeling and the probabilistic description for two agent-based
computational economic market models: the Levy-Levy-Solomon model and the
Franke-Westerhoff model. We derive time-continuous formulations of both models,
and in particular we discuss the impact of the time-scaling on the model
behavior for the Levy-Levy-Solomon model. For the Franke-Westerhoff model, we
proof that a constraint required in the original model is not necessary for
stability of the time-continuous model. It is shown that a semi-implicit
discretization of the time-continuous system preserves this unconditional
stability. In addition, this semi-implicit discretization can be computed at
cost comparable to the original model. Furthermore, we discuss possible
probabilistic descriptions of time continuous agent-based computational
economic market models. Especially, we present the potential advantages of
kinetic theory in order to derive mesoscopic desciptions of agent-based models.
Exemplified, we show two probabilistic descriptions of the Levy-Levy-Solomon
and Franke-Westerhoff model.
