Asymptotic stability of neutral-type stochastic functional differential equations; Collaborative research visit to China
Lead Research Organisation:
University of Strathclyde
Department Name: Statistics and Modelling Science
Abstract
The proposed collaboration is mainly to study the asymptotic stability of neutral-type stochastic functional differential equations. Shen and the PI already have an ongoing collaboration in this area. The collaboration was initiated in 2004 when the PI visited China. The collaboration has since continued mainly via e-mails. Three papers resulting from the collaboration to date have been published or accepted for publication. In these papers, we have not only established some very interesting results on stochastic stability but also identified some new problems. For example, it is observed that the existing stability criteria on neutral-type equations require the linear growth condition and constant delay. Moreover, there is little known about the stability of numerical methods applied to neutral-type equations. To get insight on these problems, we write this proposal for a travel grant for the PI to visit Shen's laboratory in order to continueour current collaboration, namely to carry out a project with 3 aims listed above. The key benefits are:(1) to give the PI an opportunity to have face-to-face discussions with Professor Shen as well as other researchers visiting Professor Shen's Laboratory;(2) to make use of the excellent computing facility in Professor Shen's laboratory to carry out the computational side of the project.
Organisations
People |
ORCID iD |
Xuerong Mao (Principal Investigator) |
Publications
Feng L
(2016)
Mean percentage of returns for stock market linked savings accounts
in Applied Mathematics and Computation
Füllekrug M
(2012)
Energetic Charged Particles Above Thunderclouds
in Surveys in Geophysics
Guo Q
(2017)
Multi-level Monte Carlo methods with the truncated Euler-Maruyama scheme for stochastic differential equations
in International Journal of Computer Mathematics
Guo Q
(2016)
Almost Sure Exponential Stability of Stochastic Differential Delay Equations
in SIAM Journal on Control and Optimization
Hu L
(2013)
Stability and boundedness of nonlinear hybrid stochastic differential delay equations
in Systems & Control Letters
Li X
(2010)
Approximate solutions of stochastic differential delay equations with Markovian switching
in Journal of Difference Equations and Applications
Liu W
(2016)
Almost sure stability of the Euler-Maruyama method with random variable stepsize for stochastic differential equations
in Numerical Algorithms
Luo Q
(2006)
New criteria on exponential stability of neutral stochastic differential delay equations
in Systems & Control Letters
Luo Q
(2011)
Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations
in Automatica
Mao X
(2016)
Almost Sure Exponential Stabilization by Discrete-Time Stochastic Feedback Control
in IEEE Transactions on Automatic Control
Description | "Three key aims of the proposal have been achieved. The research outcomes have formed 7 research papers listed below. The main findings of these papers are described as follows: In papers (1) and (2), we establish some useful criteria on the exponential stability for neutral-type stochastic differential delay equations. These criteria no longer require the coefficients of the underlying equations to obey the linear growth condition nor the time delay to be a constant. Moreover, the key condition on the diffusion operator associated with the underlying equations takes a much more general form. Our new stability criteria not only cover many highly non-linear neutral-type stochastic differential delay equations with variable time delays but they can also be verified much more easily than the known criteria. In papers (3), (6) and (7) we develop the techniques used in papers (1) and (2) to deal with more general neutral stochastic functional differential equations with Markovian switching. Our new existence-and-uniqueness theorem enables us to consider a wider class of nonlinear equations. We establish a powerful LaSalle-type theorem on the limit sets of the solutions. We also establish several new criteria on almost surely asymptotic stability, in particular, exponential and polynomial stability. In papers (4) and (5), we develop a numerical scheme to approximate the solutions of stochastic functional differential equations with Markovian switching. We show the strong convergence of the approximate solutions to the true solutions under the very week local Lipschitz condition. Moreover, based on the scheme developed in this paper we have also carried out certain amount of numerical simulation to examine whether numerical methods can help to reveal long-time behaviours of neutral-type equations. In particular, we have found that the numerical method can be used to study the exponential stability of linear neutral-type equations and we are currently writing a new paper on it. Publications: (1) Luo, Q., Mao, X. and Shen, Y., New criteria on exponential stability of neutral stochastic differential delay equations, Systems and Control Letters 55 (2006), 826-834. (2) Shen, Y. and Mao, X., Asymptotic behaviours of stochastic differential delay equations, Asian Journal of Control 8(1) (2006), 21-27. (3) Mao, X., Shen, Y. and Yuan, C., Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching, Stochastic Processes and their Applications 118 (2008), 1385--1406. (4) Li, X., Mao, X. and Shen, Y., Approximate solutions of stochastic differential delay equations with Markovian switching, Journal of Difference Equations and Applications 16(2-3) (2010), 195-207. (5) Mao, X., Shen, Y. and Gray, A., Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations, Journal of Computational and Applied Mathematics 235 (2011), 1213-1226. (6) Luo, Q., Mao, X. and Shen, Y., Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations, Automatica 47 (2011), 2075--2081. (7) Hu, L., Mao, X. and Shen, Y., Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Systems \& Control Letters 62 (2013), 178-187." |
Description | 2009 International Conference on Scientific Computation and Differential Equations |
Form Of Engagement Activity | Scientific meeting (conference/symposium etc.) |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Participants in your research or patient groups |
Results and Impact | Invited speaker : Participation in conference : Invited talk and session chair at 2009 International Conference on Scientific Computation and Differential Equations (SciCADE09), Beijing, China, 25-29 May 2009. . 2009 International Conference on Scientific Computation and Differential Equations |
Year(s) Of Engagement Activity | 2009 |
Description | 2nd CLSS-UK North UK Life Science Symposium |
Form Of Engagement Activity | Scientific meeting (conference/symposium etc.) |
Part Of Official Scheme? | No |
Primary Audience | Participants in your research or patient groups |
Results and Impact | Invited speaker : Participation in conference : 2nd CLSS-UK North UK Life Science Symposium, Glasgow (Invited Speaker) . 2nd CLSS-UK North UK Life Science Symposium |
Year(s) Of Engagement Activity | 2008 |
Description | 3rd Meeting of the Leverhulme International Network |
Form Of Engagement Activity | Scientific meeting (conference/symposium etc.) |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Participants in your research or patient groups |
Results and Impact | Invited speaker : Participation in conference : 3rd Meeting of the Leverhulme International Network, Chester, (Invited Speaker) . |
Year(s) Of Engagement Activity | 2008 |
Description | International Conference on Scientific Computation and Differential Equations |
Form Of Engagement Activity | Scientific meeting (conference/symposium etc.) |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Participants in your research or patient groups |
Results and Impact | Invited speaker : Participation in conference : International Conference on Scientific Computation and Differential Equations (SciCADE09), Beijing, China (Invited Speaker and Session Chair) . International Conference on Scientific Computation and Differential Equations |
Year(s) Of Engagement Activity | 2008 |
Description | Stochastic Workshop, Swansea, (Invited Speaker) |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Primary Audience | |
Results and Impact | Visitor : Invited talk : Stochastic Workshop, Swansea, (Invited Speaker) . |
Year(s) Of Engagement Activity | 2008 |