A simple model for asset price bubble formation and collapse

摘要

We consider a simple stochastic differential equation for modeling bubbles insocial context. A prime example is bubbles in asset pricing, but similarmechanisms may control a range of social phenomena driven by psychologicalfactors (for example, popularity of rock groups, or a number of studentspursuing a given major). Our goal is to study the simplest possible model inwhich every term has a clear meaning and which demonstrates several keybehaviors. The main factors that enter are tendency of mean reversion to astable value, speculative social response triggered by trend following andrandom fluctuations. The interplay of these three forces may lead to bubbleformation and collapse. Numerical simulations show that the equation hasdistinct regimes depending on the values of the parameters. We perform rigorousanalysis of the weakly random regime, and study the role of change infundamentals in igniting the bubble.