  We develop several statistical tests of the determinant of the diffusion
coefficient of a stochastic differential equation, based on discrete
observations on a time interval $[0,T]$ sampled with a time step $\Delta$. Our
main contribution is to control the test Type I and Type II errors in a non
asymptotic setting, i.e. when the number of observations and the time step are
fixed. The test statistics are calculated from the process increments. In
dimension 1, the density of the test statistic is explicit. In dimension 2, the
test statistic has no explicit density but upper and lower bounds are proved.
We also propose a multiple testing procedure in dimension greater than 2. Every
test is proved to be of a given non-asymptotic level and separability
conditions to control their power are also provided. A numerical study
illustrates the properties of the tests for stochastic processes with known or
estimated drifts.
