A stochastic model for speculative bubbles

Résumé

This paper aims to provide a simple modelling of speculative bubbles and derive some quantitative properties of its dynamical evolution. Starting from a description of individual speculative behaviours, we build and study a second order Markov process, which after simple transformations can be viewed as a turning two-dimensional Gaussian process. Then, our main problem is to obtain some bounds for the persistence rate relative to the return time to a given price. In our main results, we prove with both spectral and probabilistic methods that this rate is almost proportional to the turning frequency omega of the model and provide some explicit bounds. In the continuity of this result, we build some estimators of omega and of the pseudo-period of the prices. At last, we end the paper by a proof of the quasi-stationary distribution of the process, as well as the existence of its persistence rate.