Quadratic Variation of Functionals of Two-Parameter Wiener Process

Let [W(s, t): (s, t) E R+7, R+2 = [0, co) x [0, co), be the standard twoparameter Wiener process defined on a complete probability space (Q, F, P), i.e., a Gaussian stochastic process with EW(s, t) = 0 and EW(s, t) W(s’, t’) = Min(s, s’) Min(t, t’). We shall also assume, as we may do without restricting the generality, that W(s, t; UJ) is sample path continuous, i.e., for each w, W(.; U) is a continuous function on R+“. Let F,$ , (s, t) E R+2, be the u-field generated by the random variables [W(U, 0): 0 < u ,< s, 0 < z, < t] and augmented by the P-null sets in F. In order to define the quadratic variation of a two-parameter process we need a notation for rectangles and also the notion of the increment of a process over a rectangle. Suppose (s, t) and (s’, t’) are in R+2. Ifs < s’ and t < C, ((s, t), (s’, t’)]