  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 ob- tain
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.
