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; Lauri Viitasaari

doi:10.1007/s10959-023-01272-7

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, Lauri Viitasaari^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › Scientific › peer-review

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Abstract

We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order H>12 with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the L¹ -distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to n¹^-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.

Original language	English
Pages (from-to)	721-743
Journal	JOURNAL OF THEORETICAL PROBABILITY
Volume	37
Issue number	1
Early online date	10 Jul 2023
DOIs	https://doi.org/10.1007/s10959-023-01272-7
Publication status	Published - Mar 2024
MoE publication type	A1 Journal article-refereed

Keywords

Approximation of stochastic integral
Discontinuous integrands
Fractional Brownian motions and related processes
Sharp rate of convergence

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10.1007/s10959-023-01272-7Licence: CC BY

s10959-023-01272-7Final published version, 295 KBLicence: CC BY

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

abstract = "We consider equidistant approximations of stochastic integrals driven by H{\"o}lder continuous Gaussian processes of order H>12 with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the L1 -distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to n1-2H , 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, Fractional Brownian motions and related processes, Sharp rate of convergence",

author = "Ehsan Azmoodeh and Pauliina Ilmonen and Nourhan Shafik and Tommi Sottinen and Lauri Viitasaari",

note = "Funding Information: Pauliina Ilmonen gratefully acknowledges support from the Academy of Finland, decision number 346308 (The Centre of Excellence in Randomness and Structures). Publisher Copyright: {\textcopyright} 2023, The Author(s).",

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doi = "10.1007/s10959-023-01272-7",

language = "English",

volume = "37",

pages = "721--743",

journal = "JOURNAL OF THEORETICAL PROBABILITY",

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On Sharp Rate of Convergence for Discretization of Integrals Driven by Fractional Brownian Motions and Related Processes with Discontinuous Integrands. / Azmoodeh, Ehsan; Ilmonen, Pauliina; Shafik, Nourhan et al.
In: JOURNAL OF THEORETICAL PROBABILITY, Vol. 37, No. 1, 03.2024, p. 721-743.

Research output: Contribution to journal › Article › Scientific › peer-review

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AU - Shafik, Nourhan

AU - Sottinen, Tommi

AU - Viitasaari, Lauri

N1 - Funding Information: Pauliina Ilmonen gratefully acknowledges support from the Academy of Finland, decision number 346308 (The Centre of Excellence in Randomness and Structures). Publisher Copyright: © 2023, The Author(s).

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N2 - We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order H>12 with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the L1 -distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to n1-2H , 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.

AB - We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order H>12 with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the L1 -distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to n1-2H , 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.

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KW - Discontinuous integrands

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

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FiRST Ilmonen: Finnish centre of excellence in Randomness and Structures

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

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