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.

翻译：本文基于时间区间$[0,T]$上以时间步长$\Delta$采样的离散观测数据，发展了随机微分方程扩散系数行列式的若干统计检验方法。主要贡献在于非渐近框架下（即观测次数与时间步长固定时）对检验第一类错误与第二类错误的控制。检验统计量通过过程增量计算得到：在一维情形下，检验统计量的密度函数具有显式表达式；在二维情形下，虽无显式密度函数，但推导了其上下界。针对大于二维的情形，我们进一步提出了多重检验程序。所有检验均被证明具有给定的非渐近显著性水平，并给出了控制检验效力的可分性条件。数值研究通过已知或估计漂移项的随机过程验证了所提出检验方法的性质。