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

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...

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Bibliographic Details
Published in:Stochastic processes and their applications 2024-07, Vol.173
Main Authors: Melnykova, Anna, Reynaud-Bouret, Patricia, Samson, Adeline
Format: Article
Language:English
Subjects:
Mathematics
Probability
Online Access:Get full text
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Summary: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.
ISSN:0304-4149
1879-209X
DOI:10.1016/j.spa.2024.104372