Wavelet estimation of the diffusion coefficient in time dependent diffusion models

Abstract

The estimation problem for diffusion coefficients in diffusion processes has been studied in many papers, where the diffusion coefficient function is assumed to be a 1-dimensional bounded Lipschitzian function of the state or the time only. There is no previous work for the nonparametric estimation of time-dependent diffusion models where the diffusion coefficient depends on both the state and the time. This paper introduces and studies a wavelet estimation of the time-dependent diffusion coefficient under a more general assumption that the diffusion coefficient is a linear growth Lipschitz function. Using the properties of martingale, we translate the problems in diffusion into the nonparametric regression setting and give the Lr convergence rate. A strong consistency of the estimate is established. With this result one can estimate the time-dependent diffusion coefficient using the same structure of the wavelet estimators under any equivalent probability measure. For example, in finance, the wavelet estimator is strongly consistent under the market probability measure as well as the risk neutral probability measure.