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Approximation at first and second order of $$m$$-order integrals of the fractional Brownian motion and of certain semimartingales. (English) Zbl 1063.60079
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.

##### MSC:
 60H05 Stochastic integrals 60F05 Central limit and other weak theorems 60F15 Strong limit theorems 60G15 Gaussian processes 60J65 Brownian motion
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