Georgia Institute of Technology College of Sciences

We develop a nonparametric test to check whether the underlying continuous time process is a diffusion, i.e., 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 of diffusion models, under which the null hypothesis is equivalent to a martingale difference property of the transformed processes. Then a generalized spectral derivative test is applied to check the martingale property, where the drift function is estimated via kernel regression and the diffusion function is integrated out by quadratic variation and covariation. Such a testing procedure is feasible and convenient because the infinitesimal operator of the diffusion process, unlike the transition density, has a closed-form expression of the drift and diffusion functions. The proposed test is applicable to both univariate and multivariate continuous time processes and has a N(0,1) limit distribution under the diffusion hypothesis. Simulation studies show that the proposed test has good size and all-around power against non-diffusion alternatives in finite samples. We apply the test to a number of financial time series and find some evidence against the diffusion hypothesis.