from the stochastic calculus for two-parameter martingales developed in the eighties. Let us briefly recall the context. Consider a two- parameter Wiener process $(W_{s,t})_{s,t\in [0,1]}$ and let us express the random variable $W_{s,t}^{2}$ (here $W_{s,t}$ is one dimensional random variable and $W_{s,t}^{2}$ is its square; we mention this in order to avoid the confusion with the components of the two-dimensional Brownian sheet that appears in Section \[prel\]) as a stochastic integral plus a deterministic integral in the spirit of the classical Itô formula. Let $t,s \in [0,1] $ be fixed and let $$0=t_{0}<t_{1}<\ldots t_{n}=t
\mbox{ and }
0=s_{0}<s_{1}<\ldots s_{m} = s$$ be two partitions of the intervals $[0,t]$ and $[0,s]$ respectively. For fixed $t$, we can write $$\begin{aligned}
W_{s,t}^{2}= 2\sum_{i=0}^{m-1} W_{s_{i}, t }\left(W_{s_{i+1}, t }-W_{s_{i}, t }\right)+ \sum_{i=0}^{m-1} \left( W_{s_{i+1}, t }-W_{s_{i}, t }\right) ^{2}  .\end{aligned}$$ It is standard to prove that the last summand $\sum_{i=0}^{m-1} \left( W_{s_{i+1}, t }-W_{s_{i}, t }\right) ^{2}  $ converges in $L^{2}(\Omega)$ as $m\to \infty$ to $st$ which, in some sense, constitutes the quadratic variation of the two-parameter martingale $W_{s,t}$. Let us analyze the first summand. Writing $g(t)= g(0)+ \sum_{j=0}^{n-1} (g(t_{j+1}) -g(t_{j}))$ we get $$\begin{aligned}
\sum_{i=0}^{m-1} W_{s_{i}, t }\left(W_{s_{i+1}, t }-W_{s_{i}, t }\right)
&=& \sum_{i=0}^{m-1} 