Non-asymptotic statistical test of the diffusion coefficient of stochastic differential equations

Anna Melnykova (1) , Patricia Reynaud-Bouret (2) , Adeline Samson (3)

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 UGA - Université Grenoble Alpes

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 ∆. 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.

Mots clés: Statistical tests; non-asymptotic settings; stochastic differential equations

Domaines: Probabilités [math.PR]

Anna Melnykova, Patricia Reynaud-Bouret, Adeline Samson. Non-asymptotic statistical test of the diffusion coefficient of stochastic differential equations. Stochastic Processes and their Applications, 2024, pp.104372. ⟨10.1016/j.spa.2024.104372⟩. ⟨hal-04167385v2⟩