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

TL;DR

This paper investigates how the quadratic covariation between a non-smooth transformation of Brownian motion and the original process behaves as the transformation's scale shrinks, revealing polynomial decay rates.

Contribution

It introduces a novel analysis of the asymptotic behavior of quadratic covariation for non-smooth functions of Brownian motion, including an approximation scheme based on backward-forward Itô integrals.

Findings

Quadratic covariation decays polynomially in epsilon for functions in C^α.

Established an epsilon-dependent approximation scheme for the covariation.

Provided estimates for the approximation, advancing understanding of non-smooth transformations.

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.