Existence Uniqueness and Stability of Nonlocal Neutral Stochastic Differential Equations with Random Impulses and Poisson Jumps

Year 2022, Volume: 5 Issue: 3, 250 - 262, 30.09.2022

Dimplekumar Chalishajar , Ramkumar Kumark , K. Ravikumar , Geoff Cox

https://doi.org/10.53006/rna.973653

Abstract

This manuscript aims to investigate the existence, uniqueness, and stability of non-local random impulsive
neutral stochastic differential time delay equations (NRINSDEs) with Poisson jumps. First, we prove the
existence of mild solutions to this equation using the Banach fixed point theorem. Next, we prove the
stability via continuous dependence initial value. Our study extends the work of Wang and Wu [15] where
the time delay is addressed by the prescribed phase space B (defined in Section 3). An example is given to
illustrate the theory.

Keywords

Existence, Stability, Random impulsive, Stochastic differential system, Banach fixed point theorem

Supporting Institution

n/a

Project Number

-

Thanks

We would like to thank you to the reviewers for their fruitful comments and suggestions to improve this manuscript.

References

• [1] X. Mao, Stochastic Differential Equations and Applications, M. Horwood, Chichester, (1997).
• [2] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge: Cambridge University Press, (1992).
• [3] B. Oksendal, Stochastic differential Equations: An introduction with Applications, Springer Science and Business Media, (2013).
• [4] D. Applebaum, Levy Process and Stochastic Calculus, Cambridge, UK: Cambridge University Press, (2009).
• [5] X. Yang, Q. Zhu, pth moment exponential stability of stochastic partial differential equations with Poisson jumps, Asian J. Control. 16 (2014) 1482-1491.
• [6] A. Anguraj, K. Ravikumar, Existence and stability of impulsive stochastic partial neutral functional differential equations with in?nite delays and Poisson jumps, Discontinuity, Nonlinearity, and Complexity, 9(2) (2020) 245-255.
• [7] A. Anguraj, K. Ramkumar, E. M. Elsayed, Existence, uniqueness and stability of impulsive stochastic partial neutral functional differential equations with infinite delays driven by a fractional Brownian motion, Discontinuity, Nonlinearity, and Complexity, 9(2) (2020) 327-337.
• [8] A. Anguraj, K. Karthikeyan, Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions, Nonlinear Analysis Theory Methods and Applications, 70(7) (2009) 2717-2721.
• [9] E. Heranadez, Marco Rabello, H. R. Henriquez, Existence of solutions for impulsive partial neutral functional differential equations, J. Math. Anal. Appl. 311 (2007) 1135-1158.
• [10] V. Lakshmikantham, D. D. Bianov, P. S. Simenonov, Theory of Impulsive Differential equations, World Scientific, Singapore, (1989).
• [11] A. M. Samoilenko, N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, (1995).
• [12] S. J. Wu, X. Z. Meng, Boundedness of nonlinear differential systems with impulsive efects on random moments, Acta Mathematicae Applicatae Sinica, 20(1) (2006) 147-154.
• [13] S. Li, L. Shu, X. B. Shu, F. Xu, Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays, Stochastics, 91(6) (2019) 857-872.
• [14] A. Anguraj, K. Ravikumar, J. J. Nieto, On stability of stochastic differential equations with random impulses driven by Poisson jumps, Stochastics An International Journal of Probability and Stochastic Processes, 93(5) (2021) 682-696.
• [15] T. Wang, S. Wu, Random impulsive model for stock prices and its application for insurers, Master thesis (in Chinese), Shanghai, East China Normal University, (2008).
• [16] A. Anguraj, A. Vinodkumar, Existence and uniqueness of neutral functional differential equations with random impulses, International Journal of Nonlinear Science, 8 (2009) 412-418.
• [17] A. Vinodkumar, M. Gowrisankar, P. Mohankumar, Existence, uniqueness and stability of random impulsive neutral partial di?erential equations, Journal of the Egyptian Mathematical Society, 23(1) (2015) 31-36.
• [18] Z. Li, X. B. Shu, T. Miao, The existence of solutions for SturmLiouville differential equation with random impulses and boundary value problems, Boundary Value Problems, 2021(1) (2021) 1-23.
• [19] Z. Li, X. B. Shu, F. Xu, The existence of upper and lower solutions to second order random impulsive differential equation with boundary value problem, AIMS Mathematics, 5(6) (2020) 6189-6210.
• [20] Y. Guo, X. B. Shu, Q. Yin, Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions, Discrete & Continuous Dynamical Systems-B, (2021) .
• [21] L. Shu, X. B. Shu, Q. Zhu, F. Xu, Existence and exponential stability of mild solutions for second-order neutral stochastic functional differential equation with random impulses, Journal of Applied Analysis & Computation, 11(1) (2021) 59-80.
• [22] D. Chalishajar, K. Ramkumar, K. Ravikumar, Impulsive-integral inequalities for attracting and quasi-invariant sets of neutral stochastic partial functional integrodi?erential equations with impulsive effects, J. Nonlinear Sci. Appl. 13 (2020) 284-292.