Estimation of quadratic variation for two-parameter diffusions

In this paper we give a central limit theorem for the weighted quadratic variation process of a two-parameter Brownian motion. As an application, we show that the discretized quadratic variations ∑ i = 1 [ n s ] ∑ j = 1 [ n t ] | Δ i , j Y | 2 of a two-parameter diffusion Y = ( Y ( s , t ) ) ( s , t ) ∈ [ 0 , 1 ] 2 observed on a regular grid G n form an asymptotically normal estimator of the quadratic variation of Y as n goes to infinity.