Testing whether the underlying continuous-time process follows a diffusion: An infinitesimal operator-based approach

We develop a nonparametric test to check whether a process can be represented by a stochastic differential equation driven only by a Brownian motion. Our testing procedure utilizes the infinitesimal operator-based martingale characterization combined with a generalized spectral approach. Such a testing procedure is feasible and convenient because the infinitesimal operator of the diffusion process has a closed-form expression. The proposed test is applicable to both univariate and multivariate processes and has an N(0,1) limit distribution under the diffusion hypothesis. Simulation and empirical studies show that the proposed test has reasonable performance in small samples.