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Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation

Shao-Qin Zhang, Chenggui Yuan Orcid Logo

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume: 151, Issue: 4, Pages: 1278 - 1304

Swansea University Author: Chenggui Yuan Orcid Logo

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DOI (Published version): 10.1017/prm.2020.60

Abstract

In this paper, a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>\ff 1 2$ is studied. The drift term of the equation is locally Lipschitz and unbounded in the neighborhood of the origin. The existence, uniqueness and positivi...

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Published in: Proceedings of the Royal Society of Edinburgh: Section A Mathematics
ISSN: 0308-2105 1473-7124
Published: Cambridge University Press (CUP) 2020
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URI: https://cronfa.swan.ac.uk/Record/cronfa57824
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spelling 2021-10-04T15:23:32.9734395 v2 57824 2021-09-09 Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation 22b571d1cba717a58e526805bd9abea0 0000-0003-0486-5450 Chenggui Yuan Chenggui Yuan true false 2021-09-09 SMA In this paper, a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>\ff 1 2$ is studied. The drift term of the equation is locally Lipschitz and unbounded in the neighborhood of the origin. The existence, uniqueness and positivity of the solutions are proved. We estimate moments including the negative power moments. We also develop the implicit Euler scheme, proved that the scheme is positivity preserving and strong convergent, and obtain rate of convergence. Furthermore, we show that our results can be applied to stochastic interest rate models such as mean-reverting stochastic volatility model and strongly nonlinear A\"it-Sahalia type model by using Lamperti transformation Journal Article Proceedings of the Royal Society of Edinburgh: Section A Mathematics 151 4 1278 1304 Cambridge University Press (CUP) 0308-2105 1473-7124 locally Lipschitz drift; fractional Brownian motion; implicit Euler scheme; optimal strong convergence rate; interest rate models 2 9 2020 2020-09-02 10.1017/prm.2020.60 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2021-10-04T15:23:32.9734395 2021-09-09T08:16:37.2274217 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Shao-Qin Zhang 1 Chenggui Yuan 0000-0003-0486-5450 2 57824__20798__82af802641f24c8fad50d085c4393621.pdf RIS.pdf 2021-09-09T08:20:09.9018245 Output 323446 application/pdf Accepted Manuscript true Released under the terms of a Creative Commons Attribution Non-Commercial No Derivatives License (CC-BY-NC-ND) true eng https://creativecommons.org/licenses/by-nc-nd/4.0/
title Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation
spellingShingle Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation
Chenggui Yuan
title_short Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation
title_full Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation
title_fullStr Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation
title_full_unstemmed Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation
title_sort Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation
author_id_str_mv 22b571d1cba717a58e526805bd9abea0
author_id_fullname_str_mv 22b571d1cba717a58e526805bd9abea0_***_Chenggui Yuan
author Chenggui Yuan
author2 Shao-Qin Zhang
Chenggui Yuan
format Journal article
container_title Proceedings of the Royal Society of Edinburgh: Section A Mathematics
container_volume 151
container_issue 4
container_start_page 1278
publishDate 2020
institution Swansea University
issn 0308-2105
1473-7124
doi_str_mv 10.1017/prm.2020.60
publisher Cambridge University Press (CUP)
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description In this paper, a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H>\ff 1 2$ is studied. The drift term of the equation is locally Lipschitz and unbounded in the neighborhood of the origin. The existence, uniqueness and positivity of the solutions are proved. We estimate moments including the negative power moments. We also develop the implicit Euler scheme, proved that the scheme is positivity preserving and strong convergent, and obtain rate of convergence. Furthermore, we show that our results can be applied to stochastic interest rate models such as mean-reverting stochastic volatility model and strongly nonlinear A\"it-Sahalia type model by using Lamperti transformation
published_date 2020-09-02T04:13:51Z
_version_ 1763753938548424704
score 10.999161

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