Least squares estimator for the parameter of the fractional Ornstein-Uhlenbeck sheet

Abstract

We will study the least square estimator $\hat{\theta}_{T,S}$ for the drift parameter $\theta$ of the fractional Ornstein-Uhlenbeck sheet which is defined as the solution of the Langevin equation X_{t,s}= -\theta \int^{t}_{0} \int^{s}_{0} X_{v,u}dvdu + B^{\alpha, \beta}_{t,s}, \qquad (t,s) \in [0,T]\times [0,S] driven by the fractional Brownian sheet $B^{\alpha ,\beta}$ with Hurst parameters $\alpha, \beta$ in $(1/2, 5/8)$. Using the properties of multiple Wiener-It\^o integrals we prove that the estimator is strongly consistent for the parameter $\theta$. In contrast to the one-dimensional case, the estimator $\hat{\theta}_{T,S}$ is not asymptotically normal.