Solvability and stability of stochastic singular difference equations with constant coefficient matrices of index-ν

Abstract

In this paper, we shall deal with stochastic singular difference equations (SSDEs) with constant coefficient matrices and nonlinear stochastic perturbations. The solvability and stability of SSDEs are difficult to study because of the singularity of the leading coefficient matrix. An index-ν concept is derived and formulas of solution are established for these equations. The continuous dependence of solution on initial condition is also considered. Finally, the stability of SSDEs is studied by using the method of Lyapunov functions. Some examples are given to illustrate the results.

Keywords:

• Stochastic singular difference equations
• index
• solvability
• stability
• stochastic perturbation

Acknowledgments

The first author would like to thank Vietnam Institute for Advanced Study in Mathematics (VIASM) for providing a fruitful research environment and hospitality. The authors also gratefully thank the reviewers for useful comments that led to the improvements of the paper.

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article and its supplementary materials.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Foundation for Science and Technology Development (NAFOSTED) project 101.01-2021.10.

Notes on contributors

Do Duc Thuan

Do Duc Thuan received the B.Sc., M.Sc., and Ph.D. degrees in mathematics from VNU University of Science, Hanoi, Vietnam in 2007, 2009 and 2012, respectively. He was a postdoctoral research fellow at the Institute of Mathematics, Technical University of Berlin, Germany in 2012–2013. D.D. Thuan began with Hanoi University of Science and Technology as a lecturer in 2007, where he is currently an associate professor in mathematics. His research interests include control theory, differential–algebraic equations and singular difference equations.

Nguyen Hong Son

Nguyen Hong Son has received the Ph.D. degree in mathematics from VNU University of Science, Hanoi, Vietnam and has worked at Tran Quoc Tuan University, Hanoi, Vietnam. His research interests include stochastic differential equations, stochastic difference equations, stability and robust stability of differential algebraic equations and singular difference equations with respect to stochastic perturbations.