Goodness-of-Fit Tests for Perturbed Dynamical Systems

We consider the goodness of fit testing problem for stochastic differential equation with small diffiusion coefficient. The basic hypothesis is always simple and it is described by the known trend coefficient. We propose several tests of the type of Cramer-von Mises, Kolmogorov-Smirnov and Chi-Square. The power functions of these tests we study for a special classes of close alternatives. We discuss the construction of the goodness of fit test based on the local time and the possibility of the construction of asymptotically distribution free tests in the case of composite basic hypothesis.

💡 Research Summary

The paper addresses the problem of constructing goodness‑of‑fit (GoF) tests for stochastic differential equations (SDEs) with a small diffusion coefficient. The model under study is
 dXₜ = S(Xₜ) dt + ε σ(Xₜ) dWₜ, X₀ = x₀, 0 ≤ t ≤ T,
where the diffusion term ε²σ²(·) is known and the drift (trend) function S(·) is the object of inference. The basic hypothesis H₀ is simple: S(·) = S₀(·). The alternative hypothesis allows any drift different from S₀, i.e., a non‑parametric deviation. The asymptotic regime considered is ε → 0, which corresponds to a “small‑noise” situation. Under Lipschitz conditions on S₀ and σ, the SDE has a unique strong solution and the stochastic trajectory Xₜ converges uniformly to the deterministic solution xₜ of the ordinary differential equation ẋₜ = S₀(xₜ). Moreover, the scaled deviation ε⁻¹(Xₜ − xₜ) converges in probability to the solution x^{(1)}ₜ of the linearized equation d x^{(1)}ₜ = S₀′(xₜ) x^{(1)}ₜ dt + σ(xₜ) dWₜ.

Cramér‑von Mises and Kolmogorov‑Smirnov type tests.
Two statistics are introduced:

 δ_ε = ∫₀ᵀ