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

We will study the least square estimator (theta) over cap (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,X-s = -theta integral(t)(0) integral(s)(0) X(v,u)dvdu + B-t,s(alpha,beta) , (t, s) is an element of [0, T] x [0, S]. driven by the fractional Brownian sheet B-alpha,B-beta with Hurst parameters alpha, beta in (1/2, 5/8). Using the properties of multiple Wiener-Ito integrals we prove that the estimator is strongly consistent for the parameter theta. In contrast to the one-dimensional case, the estimator (theta) over cap (T,S) is not asymptotically normal. (c) 2011 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.