On Sharp Rate of Convergence for Discretization of Integrals Driven by Fractional Brownian Motions and Related Processes with Discontinuous Integrands

Ehsan Azmoodeh (), Pauliina Ilmonen (), Nourhan Shafik (), Tommi Sottinen () and Lauri Viitasaari ()
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Ehsan Azmoodeh: University of Liverpool
Pauliina Ilmonen: Aalto University School of Science
Nourhan Shafik: Aalto University School of Science
Tommi Sottinen: University of Vaasa
Lauri Viitasaari: Uppsala University

Journal of Theoretical Probability, 2024, vol. 37, issue 1, 721-743

Abstract: Abstract We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order $$H>\frac{1}{2}$$ H > 1 2 with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the $$L^1$$ L 1 -distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to $$n^{1-2H}$$ n 1 - 2 H , which is twice as good as the best known results in the case of discontinuous integrands and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply a change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.

Keywords: Approximation of stochastic integral; Discontinuous integrands; Sharp rate of convergence; Fractional Brownian motions and related processes; 60G15; 60G22; 60H05 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-023-01272-7

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