Évènements

We develop a statistical test on the determinant of the diffusion coefficient in a 1 or 2-dimensional stochastic differential equations from discrete observations on a fixed time interval [0,T] sampled with a fixed time step.
We propose a test statistic based on increments of the process which guarantees the control of the level of the test in a non asymptotic setting. In dimension 1, the test density is known explicitly even when the drift is estimated. We construct the test and give conditions under which the Type I and Type II errors can be controlled. In dimension 2, the test statistic has not an explicit density but upper and lower bounds are provided. We then give conditions under which the Type I and Type II errors of the test procedure can be controlled. A numerical study illustrates the properties of the tests for stochastic processes with known or unknown drifts.

Among all positional numeration systems, the widely studied Bertrand numeration systems are defined by a simple criterion in terms of their numeration languages. In 1989, Bertrand-Mathis characterized them via representations in a real base $\beta$. However, the given condition turns out to be not necessary. In this talk, I will present a correction of Bertrand-Mathis’ result. The main difference arises when $\beta$ is a simple Parry number, in which case two associated Bertrand numeration systems are derived. Along the way, we define a non-canonical $\beta$-shift and study its properties analogously to those of the usual canonical one.