A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion

SUO, YONGQIANG

doi:10.23889/SUthesis.57330

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A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion / YONGQIANG SUO

Swansea University Author: YONGQIANG SUO

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DOI (Published version): 10.23889/SUthesis.57330

Abstract

In this thesis, we mainly study some properties for certain stochastic di↵er-ential equations.The types of stochastic di↵erential equations we are interested in are (i) stochastic di↵erential equations driven by Brownian motion, (ii) stochastic functional di↵erential equations driven by fractional B...

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Published:	Swansea 2021
Institution:	Swansea University
Degree level:	Doctoral
Degree name:	Ph.D
Supervisor:	Yuan, Chenggui ; Neate, Andrew
URI:	https://cronfa.swan.ac.uk/Record/cronfa57330
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first_indexed	2021-07-15T10:53:12Z
last_indexed	2021-07-16T03:23:57Z
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spelling	2021-07-15T12:03:31.3810575 v2 57330 2021-07-15 A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion 9f5e288c171f1e0f01062b8a5a9007af YONGQIANG SUO YONGQIANG SUO true false 2021-07-15 In this thesis, we mainly study some properties for certain stochastic di↵er-ential equations.The types of stochastic di↵erential equations we are interested in are (i) stochastic di↵erential equations driven by Brownian motion, (ii) stochastic functional di↵erential equations driven by fractional Brownian motion, (iii) McKean-Vlasov stochastic di↵erential equations driven by Brownian motion,(iv) McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion.The properties we investigate include the weak approximation rate of Euler-Maruyama scheme, the central limit theorem and moderate deviation principle for McKean-Vlasov stochastic di↵erential equations. Additionally, we investigate the existence and uniqueness of solution to McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion, and then the Bismut formula of Lion’s derivatives for this model is also obtained.The crucial method we utilised to establish the weak approximation rate of Euler-Maruyama scheme for stochastic equations with irregular drift is the Girsanov transformation. More precisely, giving a reference stochastic equa-tions, we construct the equivalent expressions between the aim stochastic equations and associated numerical stochastic equations in another proba-bility spaces in view of the Girsanov theorem.For the Mckean-Vlasov stochastic di↵erential equation model, we ﬁrst construct the moderate deviation principle for the law of the approxima-tion stochastic di↵erential equation in view of the weak convergence method. Subsequently, we show that the approximation stochastic equations and the McKean-Vlasov stochastic di↵erential equations are in the same exponen-tially equivalent family, and then we establish the moderate deviation prin-ciple for this model.Based on the result of Well-posedness for Mckean-Vlasov stochastic di↵er-ential equation driven by fractional Brownian motion, by using the Malliavin analysis, we ﬁrst establish a general result of the Bismut type formula for Lions derivative, and then we apply this result to the non-degenerate case of this model. E-Thesis Swansea Weak approximation, moderate deviation principle, central limit theorem, Lions derivative, Bismut formula 12 7 2021 2021-07-12 10.23889/SUthesis.57330 COLLEGE NANME COLLEGE CODE Swansea University Yuan, Chenggui ; Neate, Andrew Doctoral Ph.D 2021-07-15T12:03:31.3810575 2021-07-15T11:49:45.7123543 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics YONGQIANG SUO 1 57330__20412__e7bb62a1e2b14161a59dc2e502d5255e.pdf Suo_Yonggiang_PhD_Thesis_Final_Redacted_Signature.pdf 2021-07-15T11:59:59.4073279 Output 1627633 application/pdf E-Thesis – open access true Copyright: The author, Yongqiang Suo, 2021. true eng
title	A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion
spellingShingle	A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion YONGQIANG SUO
title_short	A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion
title_full	A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion
title_fullStr	A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion
title_full_unstemmed	A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion
title_sort	A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion
author_id_str_mv	9f5e288c171f1e0f01062b8a5a9007af
author_id_fullname_str_mv	9f5e288c171f1e0f01062b8a5a9007af_***_YONGQIANG SUO
author	YONGQIANG SUO
author2	YONGQIANG SUO
format	E-Thesis
publishDate	2021
institution	Swansea University
doi_str_mv	10.23889/SUthesis.57330
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 thesis, we mainly study some properties for certain stochastic di↵er-ential equations.The types of stochastic di↵erential equations we are interested in are (i) stochastic di↵erential equations driven by Brownian motion, (ii) stochastic functional di↵erential equations driven by fractional Brownian motion, (iii) McKean-Vlasov stochastic di↵erential equations driven by Brownian motion,(iv) McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion.The properties we investigate include the weak approximation rate of Euler-Maruyama scheme, the central limit theorem and moderate deviation principle for McKean-Vlasov stochastic di↵erential equations. Additionally, we investigate the existence and uniqueness of solution to McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion, and then the Bismut formula of Lion’s derivatives for this model is also obtained.The crucial method we utilised to establish the weak approximation rate of Euler-Maruyama scheme for stochastic equations with irregular drift is the Girsanov transformation. More precisely, giving a reference stochastic equa-tions, we construct the equivalent expressions between the aim stochastic equations and associated numerical stochastic equations in another proba-bility spaces in view of the Girsanov theorem.For the Mckean-Vlasov stochastic di↵erential equation model, we ﬁrst construct the moderate deviation principle for the law of the approxima-tion stochastic di↵erential equation in view of the weak convergence method. Subsequently, we show that the approximation stochastic equations and the McKean-Vlasov stochastic di↵erential equations are in the same exponen-tially equivalent family, and then we establish the moderate deviation prin-ciple for this model.Based on the result of Well-posedness for Mckean-Vlasov stochastic di↵er-ential equation driven by fractional Brownian motion, by using the Malliavin analysis, we ﬁrst establish a general result of the Bismut type formula for Lions derivative, and then we apply this result to the non-degenerate case of this model.
published_date	2021-07-12T04:12:59Z
_version_	1763753884247916544
score	11.016258

A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion / YONGQIANG SUO

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