  We introduce and study a non-equilibrium continuous-time dynamical model of
the price of a single asset traded by a population of heterogeneous interacting
agents in the presence of uncertainty and regulatory constraints. The model
takes into account (i) the price formation delay between decision and
investment by the second-order nature of the dynamical equations, (ii) the
linear and nonlinear mean-reversal or their contrarian in the form of
speculative price trading, (iii) market friction, (iv) uncertainty in the
fundamental value which controls the amplitude of mispricing, (v) nonlinear
speculative momentum effects and (vi) market regulations that may limit large
mispricing drifts. We find markets with coexisting equilibrium, conventions and
business cycles, which depend on (a) the relative strength of value-investing
versus momentum-investing, (b) the level of uncertainty on the fundamental
value and (c) the degree of market regulation. The stochastic dynamics is
characterized by nonlinear geometric random walk-like processes with
spontaneous regime shifts between different conventions or business cycles.
This model provides a natural dynamical framework to model regime shifts
between different market phases that may result from the interplay between the
effects (i-vi).
