A stochastic model for speculative bubbles

Sébastien Gadat (1) , Laurent Miclo (1) , Fabien Panloup (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

The aim of this paper is to provide a simple modelling of speculative bubbles and to derive some quantitative properties of its dynamical evolution. Starting from a description of individual speculative behaviours, we build and study a second order non-reversible 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 pseudoperiod of the prices. At last, we prove the existence of the persistence rate and of the associated quasi-stationary distribution.

Mots clés: Statistics of processes; Quasistationary Distribution; Statistics of processes.; Speculative bubble; Persistence rate; Gaussian Process; Diffusion Bridge

Domaines: Probabilités [math.PR]

Sébastien Gadat, Laurent Miclo, Fabien Panloup. A stochastic model for speculative bubbles. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2015, 12, pp.491-532. ⟨hal-00937447⟩