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

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

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon
2 SVH - Statistique pour le Vivant et l’Homme
3 LJAD - Laboratoire Jean Alexandre Dieudonné

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.

Keywords: Statistical tests; non-asymptotic settings; stochastic differential equations

Domains: Probability [math.PR]

Anna Melnykova, Adeline Leclercq-Samson, Patricia Reynaud-Bouret. Non-asymptotic statistical test of the diffusion coefficient of stochastic differential equations. 2023. ⟨hal-04167385v2⟩