Quadratic covariation estimates in non-smooth stochastic calculus

Given a Brownian Motion WW, in this paper we study the asymptotic behavior, as ε→0ε→0, of the quadratic covariation between f(εW)f(εW) and WW in the case in which ff is not smooth. Among the main features discovered is that the speed of the decay in the case f∈Cαf∈Cα is at least polynomial in εε and not exponential as expected. We use a recent representation as a backward–forward Itô integral of [f(εW),W][f(εW),W] to prove an εε-dependent approximation scheme which is of independent interest. We get the result by providing estimates to this approximation. The results are then adapted and applied to generalize the results of Almada Monter and Bakhtin (2011) and Bakhtin (2011) related to the small noise exit from a domain problem for the saddle case.