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

TL;DR

This paper introduces non-asymptotic statistical tests for the diffusion coefficient of stochastic differential equations, controlling error rates with fixed sample sizes and providing explicit or bounded densities for test statistics.

Contribution

It develops novel non-asymptotic tests for the diffusion coefficient, including explicit density in 1D, bounds in 2D, and a multiple testing procedure for higher dimensions.

Findings

Tests control Type I and II errors at fixed sample sizes.

Explicit density for 1D test statistic; bounds for 2D.

Numerical validation with stochastic processes.

Abstract

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.