Approximation at first and second order of {\(m\)}-order integrals of the fractional {B}rownian motion and of certain semimartingales
The authors investigate the accurate convergence of some approximations of \(m\)-order integrals which appear when one performs stochastic calculus with respect to processes which are not semimartingales, for instance the fractional Brownian motion. Let \(X\) be the fractional Brownian motion of any Hurst index \(H\) in \((0,1)\) (resp. a semimartingale). Let \(Y\) be a continuous process and let \(m\) be a positive integer. The authors study the existence of the limit of the approximation of \(m\)-order integral of \(Y\) with respect to \(X\) and prove that the limits are almost sure, uniformly on each compact interval, and are in terms of the \(m\)th moment of the Gaussian standard random variable. In particular, if \(m\) is an odd integer, the limit equals to zero. They also show that the limit is a Brownian motion when \(X\) is the fractional Brownian motion of index \(H\) in \((0,1]\), and it is in terms of a two-dimensional standard Brownian motion when \(X\) is a semimartingale.
scientific article; zbMATH DE number 2142122