Rate of Convergence for Discretization of Integrals with Respect to Fractional Brownian Motion

Ehsan Azmoodeh () and Lauri Viitasaari ()
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Ehsan Azmoodeh: University of Luxembourg
Lauri Viitasaari: Aalto University School of Science

Journal of Theoretical Probability, 2015, vol. 28, issue 1, 396-422

Abstract: Abstract In this article, a uniform discretization of stochastic integrals $$\int _{0}^{1} f^{\prime }_-(B_t)\mathrm d B_t$$ ∫ 0 1 f − ′ ( B t ) d B t , where $$B$$ B denotes the fractional Brownian motion with Hurst parameter $$H \in (\frac{1}{2},1)$$ H ∈ ( 1 2 , 1 ) , is considered for a large class of convex functions $$f$$ f . In Azmoodeh et al. (Stat Decis 27:129–143, 2010), for any convex function $$f$$ f , the almost sure convergence of uniform discretization to such stochastic integral is proved. Here, we prove $$L^r$$ L r -convergence of uniform discretization to stochastic integral. In addition, we obtain a rate of convergence. It turns out that the rate of convergence can be brought arbitrarily close to $$H - \frac{1}{2}$$ H − 1 2 .

Keywords: Fractional Brownian motion; Stochastic integral; Discretization; Rate of convergence; 60G22; 60H05; 41A25 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-013-0495-y

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