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

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