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

Abstract

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 $W$ 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 $\eps$ and not exponential as expected. We use a recent representation as a backward- forward It\^o integral of $[f (\eps W), W]$ to prove an $\eps$-dependent approximation scheme which is of independent interest. We get the result by providing estimates to this approximation.