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Density Estimates for the Solutions of Backward Stochastic Differential Equations Driven by Gaussian Processes

Xiliang Fan, Jiang-lun Wu

Potential Analysis, Volume: 54, Issue: 3, Pages: 483 - 501

Swansea University Author: Jiang-lun Wu

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DOI (Published version): 10.1007/s11118-020-09835-7

Abstract

The aim of this paper is twofold. Firstly, we derive upper and lower non- Gaussian bounds for the densities of the marginal laws of the solutions to backward stochas- tic differential equations (BSDEs) driven by fractional Brownian motions. Our arguments consist of utilising a relationship between f...

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Published in: Potential Analysis
ISSN: 0926-2601 1572-929X
Published: Springer Science and Business Media LLC 2021
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URI: https://cronfa.swan.ac.uk/Record/cronfa53408
first_indexed 2020-02-03T13:24:56Z
last_indexed 2025-03-21T09:44:06Z
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spelling 2025-03-20T14:56:31.7631082 v2 53408 2020-02-03 Density Estimates for the Solutions of Backward Stochastic Differential Equations Driven by Gaussian Processes dbd67e30d59b0f32592b15b5705af885 Jiang-lun Wu Jiang-lun Wu true false 2020-02-03 The aim of this paper is twofold. Firstly, we derive upper and lower non- Gaussian bounds for the densities of the marginal laws of the solutions to backward stochas- tic differential equations (BSDEs) driven by fractional Brownian motions. Our arguments consist of utilising a relationship between fractional BSDEs and quasilinear partial differ- ential equations of mixed type, together with the profound Nourdin-Viens formula. In the linear case, upper and lower Gaussian bounds for the densities and the tail probabilities of solutions are obtained with simple arguments by their explicit expressions in terms of the quasi-conditional expectation. Secondly, we are concerned with Gaussian estimates for the densities of a BSDE driven by a Gaussian process in the manner that the solution can be established via an auxiliary BSDE driven by a Brownian motion. Using the transfer theorem we succeed in deriving Gaussian estimates for the solutions. Journal Article Potential Analysis 54 3 483 501 Springer Science and Business Media LLC 0926-2601 1572-929X Backward stochastic differential equations; Gaussian processes; fractional Brownian motion; density estimate; Malliavin calculus. 1 3 2021 2021-03-01 10.1007/s11118-020-09835-7 COLLEGE NANME COLLEGE CODE Swansea University Not Required The research of X. Fan was supported in part by the National Natural Science Foundation of China (Grant No. 11501009, 11871076), the Natural Science Foundation of Anhui Province (Grant No. 1508085QA03, 1908085MA07). 2025-03-20T14:56:31.7631082 2020-02-03T08:24:44.1835623 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Xiliang Fan 1 Jiang-lun Wu 2 53408__16505__faee133a03b2404588e6299a268e215d.pdf FanWu-acceptedVersion.pdf 2020-02-03T08:27:29.0865625 Output 344473 application/pdf Accepted Manuscript true 2021-02-28T00:00:00.0000000 true English
title Density Estimates for the Solutions of Backward Stochastic Differential Equations Driven by Gaussian Processes
spellingShingle Density Estimates for the Solutions of Backward Stochastic Differential Equations Driven by Gaussian Processes
Jiang-lun Wu
title_short Density Estimates for the Solutions of Backward Stochastic Differential Equations Driven by Gaussian Processes
title_full Density Estimates for the Solutions of Backward Stochastic Differential Equations Driven by Gaussian Processes
title_fullStr Density Estimates for the Solutions of Backward Stochastic Differential Equations Driven by Gaussian Processes
title_full_unstemmed Density Estimates for the Solutions of Backward Stochastic Differential Equations Driven by Gaussian Processes
title_sort Density Estimates for the Solutions of Backward Stochastic Differential Equations Driven by Gaussian Processes
author_id_str_mv dbd67e30d59b0f32592b15b5705af885
author_id_fullname_str_mv dbd67e30d59b0f32592b15b5705af885_***_Jiang-lun Wu
author Jiang-lun Wu
author2 Xiliang Fan
Jiang-lun Wu
format Journal article
container_title Potential Analysis
container_volume 54
container_issue 3
container_start_page 483
publishDate 2021
institution Swansea University
issn 0926-2601
1572-929X
doi_str_mv 10.1007/s11118-020-09835-7
publisher Springer Science and Business Media LLC
college_str Faculty of Science and Engineering
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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 The aim of this paper is twofold. Firstly, we derive upper and lower non- Gaussian bounds for the densities of the marginal laws of the solutions to backward stochas- tic differential equations (BSDEs) driven by fractional Brownian motions. Our arguments consist of utilising a relationship between fractional BSDEs and quasilinear partial differ- ential equations of mixed type, together with the profound Nourdin-Viens formula. In the linear case, upper and lower Gaussian bounds for the densities and the tail probabilities of solutions are obtained with simple arguments by their explicit expressions in terms of the quasi-conditional expectation. Secondly, we are concerned with Gaussian estimates for the densities of a BSDE driven by a Gaussian process in the manner that the solution can be established via an auxiliary BSDE driven by a Brownian motion. Using the transfer theorem we succeed in deriving Gaussian estimates for the solutions.
published_date 2021-03-01T07:41:58Z
_version_ 1829993414392807424
score 11.058331

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