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Discretizing the fractional Lévy area. (English) Zbl 1185.60076
Summary: We give sharp bounds for the Euler discretization of the Lévy area associated to a $$d$$-dimensional fractional Brownian motion. We show that there are three different regimes for the exact root mean square convergence rate of the Euler scheme, depending on the Hurst parameter $$H\in (1/4,1)$$. For $$H<3/4$$ the exact convergence rate is $$n^{-2H+1/2}$$, where $$n$$ denotes the number of the discretization subintervals, while for $$H=3/4$$ it is $$n^{-1}\sqrt{\log(n)}$$ and for $$H>3/4$$ the exact rate is $$n-1$$. Moreover, we also show that a trapezoidal scheme converges (at least) with the rate $$n-2H+1/2$$. Finally, we derive the asymptotic error distribution of the Euler scheme. For $$H\leq 3/4$$ one obtains a Gaussian limit, while for $$H>3/4$$ the limit distribution is of Rosenblatt type.

##### MSC:
 60H35 Computational methods for stochastic equations (aspects of stochastic analysis) 60H07 Stochastic calculus of variations and the Malliavin calculus 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 65C30 Numerical solutions to stochastic differential and integral equations
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