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

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 $Δ$. 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.

Figures (2)

• Figure 1: Power functions of the test of $H_0: \sigma^2 = 0.1^2$ against $H_1: \sigma^2 >0.1^2$ as a function of $\sigma_{20}^2$. Processes $X_\sigma$ are simulated for $\sigma_{20}^2$ varying between $0$ and $0.36$. Three tests are considered: the one-dimensional non-centered test $S$ with known drift (Section \ref{['sec:noncentered']}) in dashed blue line, with centered statistic $\dot S$ (Section \ref{['sec:centereddriftknown']}) in dotted green line, with centered statistics and estimated drift $\tilde{S}$ (Section \ref{['sec:centereddriftunknown']}) in plain red line. Three designs are considered: $\Delta=0.01, T=1$ (left), $\Delta=0.1, T=1$ (middle) and $\Delta=0.1, T=10$ (right).
• Figure 2: Power functions of the test of $H_0: \sigma_1^2\sigma_2^2 = \sigma_{20}^2$ against $H_1: \sigma_1^2\sigma_2^2 > \sigma_{20}^2$ as a function of $\sigma_{20}^2$. Processes $X_\sigma$ are simulated for $\sigma_{20}^2$ varying between $0$ and $0.36$. Four tests are considered: the 2-dimensional test with known drift (Section \ref{['sec:2dControlErrors']}) in plain red line, with estimated drift (Section \ref{['sec:2d_unknown_drift']}) with dotted green line, the multiple testing procedure (Section \ref{['sec:multiple_tests']}) with known drift in blue dot-dashed line and estimated drift in magenta dashed line. Three designs are considered: $\Delta=0.01, T=1$ (left), $\Delta=0.1, T=1$ (middle) and $\Delta=0.1, T=10$ (right).

Theorems & Definitions (29)

• Lemma 1
• Remark
• Proposition 1
• proof
• Proposition 2: 1d-Test with centered statistics and known drift
• proof
• Lemma 2
• Lemma 3
• Proposition 3: 1d-Test with centered statistics and unknown drift
• Remark