Asymptotic properties for the parameter estimation in stochastic (functional) differential equations with Hölder drift

Abstract

In this paper, by taking Zvonkin's transformation, we investigate parameter estimation for a class of multidimensional stochastic differential equations with small perturbation parameters in diffusion coefficients, where the drift coefficients not only have an unknown parameter θ but also are Hölder continuous. These processes may enhance the applicability of our results to considerable practical models. Due to the irregular drift, the primary challenge is dealing with the mean square error between an accurate and numerical solution. Under these settings, we demonstrate the consistency and asymptotic normality of error concerning the least squares estimator in probability when stepsize δ → 0 and small parameter ε → 0 simultaneously. Moreover, we extend the results to the case of stochastic functional differential equations.