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

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