On Simpson's rule and fractional Brownian motion with H = 1/10

TL;DR

This paper investigates the convergence properties of Simpson's rule for stochastic integrals with respect to fractional Brownian motion when H=1/10, establishing convergence in distribution and an Itô-like formula.

Contribution

It proves that Simpson's rule sums converge in distribution for H=1/10, extending previous results that required H > 1/10, and derives an Itô-like formula for this case.

Findings

Simpson's rule sums converge in distribution at H=1/10

An Itô-like formula is established for the stochastic integral

Convergence in probability holds for H > 1/10, but only in distribution at H=1/10

Abstract

We consider stochastic integration with respect to fractional Brownian motion (fBm) with $H < 1/2$. The integral is constructed as the limit, where it exists, of a sequence of Riemann sums. A theorem by Gradinaru, Nourdin, Russo & Vallois (2005) holds that a sequence of Simpson's rule Riemann sums converges in probability for a sufficiently smooth integrand $f$ and when the stochastic process is fBm with $H > 1/10$. For the case $H = 1/10$, we prove that the sequence of sums converges in distribution. Consequently, we have an It\^o-like formula for the resulting stochastic integral. The convergence in distribution follows from a Malliavin calculus theorem that first appeared in Nourdin and Nualart (2010).