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^*

^*Tämän työn vastaava kirjoittaja

University of Liverpool
University of Vaasa
Uppsala University

Tutkimustuotos: Lehtiartikkeli › Article › Scientific › vertaisarvioitu

1 Sitaatiot (Scopus)

56 Lataukset (Pure)

Abstrakti

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.

Alkuperäiskieli	Englanti
Sivut	721-743
Julkaisu	Journal of Theoretical Probability
Vuosikerta	37
Numero	1
Varhainen verkossa julkaisun päivämäärä	10 heinäk. 2023
DOI - pysyväislinkit	https://doi.org/10.1007/s10959-023-01272-7
Tila	Julkaistu - maalisk. 2024
OKM-julkaisutyyppi	A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä

Rahoitus

Pauliina Ilmonen gratefully acknowledges support from the Academy of Finland, decision number 346308 (The Centre of Excellence in Randomness and Structures).

Pääsy asiakirjaan

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

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

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1 Päättynyt

FiRST Ilmonen: Finnish centre of excellence in Randomness and Structures
Kytölä, K. (Co-PI), Ilmonen, P. (Vastuullinen johtaja), Avela, A. (Projektin jäsen), Zinetti, F. (Projektin jäsen), Laurikkala, M. (Projektin jäsen), Vesselinova, N. (Projektin jäsen) & Barrera Vargas, G. (Projektin jäsen)
01/01/2022 → 31/12/2024
Projekti: RCF Centre of Excellence

Siteeraa tätä

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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).",

year = "2024",

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

language = "English",

volume = "37",

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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.
julkaisussa: Journal of Theoretical Probability, Vuosikerta 37, Nro 1, 03.2024, s. 721-743.

Tutkimustuotos: Lehtiartikkeli › Article › Scientific › vertaisarvioitu

TY - JOUR

T1 - On Sharp Rate of Convergence for Discretization of Integrals Driven by Fractional Brownian Motions and Related Processes with Discontinuous Integrands

AU - Azmoodeh, Ehsan

AU - Ilmonen, Pauliina

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).

PY - 2024/3

Y1 - 2024/3

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.

KW - Approximation of stochastic integral

KW - Discontinuous integrands

KW - Fractional Brownian motions and related processes

KW - Sharp rate of convergence

UR - https://www.scopus.com/pages/publications/85164452966

U2 - 10.1007/s10959-023-01272-7

DO - 10.1007/s10959-023-01272-7

M3 - Article

AN - SCOPUS:85164452966

SN - 0894-9840

VL - 37

SP - 721

EP - 743

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

IS - 1

ER -

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

Abstrakti

Rahoitus

Pääsy asiakirjaan

Muut tiedostot ja linkit

Sormenjälki

Projektit

FiRST Ilmonen: Finnish centre of excellence in Randomness and Structures

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