The weak Stratonovich integral with respect to fractional Brownian motion with Hurst parameter 1/6

摘要

Let $B$ be a fractional Brownian motion with Hurst parameter $H=1/6$. It is known that the symmetric Stratonovich-style Riemann sums for $int!g(B(s)),dB(s)$ do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of càdlàg functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary It? integral with respect to a Brownian motion that is independent of $B$.