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Given a Brownian Motion $W$, in this paper we study the asymptotic behavior,as $\eps \to 0$, of the quadratic covariation between $f (\eps W)$ and $W$ inthe case in which $f$ is not smooth. Among the main features discovered is thatthe speed of the decay in the case $f \in C^\alpha$ is polynomial in $\eps$ andnot exponential as expected. We use a recent representation as a backward-forward It\^o integral of $[f (\eps W), W]$ to prove an $\eps$-dependentapproximation scheme which is of independent interest. We get the result byproviding estimates to this approximation.

Small Noise Estimates of the Quadratic Covariation between Non-smooth Transformations of Brownian Motion