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On a generalized population dynamics equation with environmental noise

Rongrong Tian, Jinlong Wei, Jiang-lun Wu Orcid Logo

Statistics & Probability Letters, Volume: 168, Start page: 108944

Swansea University Author: Jiang-lun Wu Orcid Logo

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DOI (Published version): 10.1016/j.spl.2020.108944

Abstract

We establish the existence and uniqueness of global (in time) positive strong solutions for a generalized population dynamics equation with environmental noise, while the global existence fails for the deterministic equation. Particularly, we prove the global existence of positive strong solutions f...

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Published in: Statistics & Probability Letters
ISSN: 0167-7152
Published: ELSEVIER Elsevier BV 2021
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URI: https://cronfa.swan.ac.uk/Record/cronfa55191
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spelling 2020-11-06T15:03:37.3413040 v2 55191 2020-09-16 On a generalized population dynamics equation with environmental noise dbd67e30d59b0f32592b15b5705af885 0000-0003-4568-7013 Jiang-lun Wu Jiang-lun Wu true false 2020-09-16 SMA We establish the existence and uniqueness of global (in time) positive strong solutions for a generalized population dynamics equation with environmental noise, while the global existence fails for the deterministic equation. Particularly, we prove the global existence of positive strong solutions for the followingstochastic differential equation\begin{eqnarray*}dX_t=(\theta X_t^{m_0}+kX_t^m)dt+\varepsilon X_t^{\frac{m+1}{2}}\varphi(X_t)dW_t, \ t>0, \ X_t>0,\ m> m_0\geq 1,\ \X_0=x>0,\end{eqnarray*}with $\theta,k,\varepsilon\in \mR$ being constants and $\varphi(r)=r^{\vartheta}$ or $|\log(r)|^{\vartheta} \ (\vartheta>0)$, and we also show that the index $\vartheta>0$ is sharp in the sense that if $\vartheta=0$, one can choose certain proper constants $\theta,k$ and $\varepsilon$ such that the solution $X_t$ will explode in a finite time almost surely. Journal Article Statistics & Probability Letters 168 108944 Elsevier BV ELSEVIER 0167-7152 Stochastic differential equation; Environmental noise; Explosion; Positive strong solution. 1 1 2021 2021-01-01 10.1016/j.spl.2020.108944 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University 2020-11-06T15:03:37.3413040 2020-09-16T22:37:08.0579956 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Rongrong Tian 1 Jinlong Wei 2 Jiang-lun Wu 0000-0003-4568-7013 3 55191__18185__107bfe3ebc264d26bda3f76aa63d813f.pdf TWWu.pdf 2020-09-16T23:01:46.0118025 Output 251798 application/pdf Accepted Manuscript true 2021-09-19T00:00:00.0000000 ©2020 All rights reserved. All article content, except where otherwise noted, is licensed under 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 On a generalized population dynamics equation with environmental noise
spellingShingle On a generalized population dynamics equation with environmental noise
Jiang-lun Wu
title_short On a generalized population dynamics equation with environmental noise
title_full On a generalized population dynamics equation with environmental noise
title_fullStr On a generalized population dynamics equation with environmental noise
title_full_unstemmed On a generalized population dynamics equation with environmental noise
title_sort On a generalized population dynamics equation with environmental noise
author_id_str_mv dbd67e30d59b0f32592b15b5705af885
author_id_fullname_str_mv dbd67e30d59b0f32592b15b5705af885_***_Jiang-lun Wu
author Jiang-lun Wu
author2 Rongrong Tian
Jinlong Wei
Jiang-lun Wu
format Journal article
container_title Statistics & Probability Letters
container_volume 168
container_start_page 108944
publishDate 2021
institution Swansea University
issn 0167-7152
doi_str_mv 10.1016/j.spl.2020.108944
publisher Elsevier BV
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 We establish the existence and uniqueness of global (in time) positive strong solutions for a generalized population dynamics equation with environmental noise, while the global existence fails for the deterministic equation. Particularly, we prove the global existence of positive strong solutions for the followingstochastic differential equation\begin{eqnarray*}dX_t=(\theta X_t^{m_0}+kX_t^m)dt+\varepsilon X_t^{\frac{m+1}{2}}\varphi(X_t)dW_t, \ t>0, \ X_t>0,\ m> m_0\geq 1,\ \X_0=x>0,\end{eqnarray*}with $\theta,k,\varepsilon\in \mR$ being constants and $\varphi(r)=r^{\vartheta}$ or $|\log(r)|^{\vartheta} \ (\vartheta>0)$, and we also show that the index $\vartheta>0$ is sharp in the sense that if $\vartheta=0$, one can choose certain proper constants $\theta,k$ and $\varepsilon$ such that the solution $X_t$ will explode in a finite time almost surely.
published_date 2021-01-01T04:09:13Z
_version_ 1763753647365160960
score 11.016235

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