Journal of Applied and Pure Mathematics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지
DOI QR코드

DOI QR Code

EXISTENCE UNIQUENESS AND STABILITY OF NONLOCAL NEUTRAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH RANDOM IMPULSES AND POISSON JUMPS

  • CHALISHAJAR, DIMPLEKUMAR (Department of Applied Mathematics, Mallory Hall, Virginia Military Institute) ;
  • RAMKUMAR, K. (Department of Mathematics, PSG College of Arts and Science) ;
  • RAVIKUMAR, K. (Department of Mathematics, PSG College of Arts and Science) ;
  • COX, EOFF (Department of Applied Mathematics, Mallory Hall, Virginia Military Institute)
  • Received : 2022.01.22
  • Accepted : 2022.05.24
  • Published : 2022.07.30
Download PDF
⟨ Previous Next ⟩

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 demonstrate the stability via continuous dependence initial value. Our study extends the work of Wang, and Wu [16] where the time delay is addressed by the prescribed phase space 𝓑 (defined in Section 3). To illustrate the theory, we also provide an example of our methods. Using our results, one could investigate the controllability of random impulsive neutral stochastic differential equations with finite/infinite states. Moreover, one could extend this study to analyze the controllability of fractional-order of NRINSDEs with Poisson jumps as well.

Keywords

References

  1. D. Applebaum, Levy Process and Stochastic Calculus, Cambridge University Press, Cambridge, UK, 2009.
  2. A. Anguraj and K. Ravikumar, Existence and stability of impulsive stochastic partial neutral functional differential equations with infinite delays and Poisson jumps, Discontinuity, Nonlinearity, and Complexity 9 (2020), 245-255. https://doi.org/10.5890/DNC.2020.06.006
  3. A. Anguraj, K. Ramkumar and 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 (2020), 327-337. https://doi.org/10.5890/DNC.2020.06.012
  4. A. Anguraj, K. Ravikumar and 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 (2021), 682-696. https://doi.org/10.1080/17442508.2020.1783264
  5. A. Anguraj and K. Karthikeyan, Existence of solutions for impulsive neutral functional differential equations with nonlocal conditions, Nonlinear Analysis Theory Methods and Applications 70 (2009), 2717-2721. https://doi.org/10.1016/j.na.2008.03.059
  6. A. Anguraj and A. Vinodkumar, Existence and uniqueness of neutral functional differential equations with random impulses, International Journal of Nonlinear Science 8 (2009), 412-418.
  7. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
  8. C. Dimplekumar, K. Ramkumar and K. Ravikumar, Impulsive-integral inequalities for attracting and quasi-invariant sets of neutral stochastic partial functional integrodifferential equations with impulsive effects, J. Nonlinear Sci. Appl. 13 (2020), 284-292.
  9. E. Heranadez, M. Rabello and 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 and P.S. Simenonov, Theory of Impulsive Differential equations, World Scientific, Singapore, 1989.
  11. S. Li, L. Shu, X.B. Shu and F. Xu, Existence and Hyers-Ulam stability of random impulsive stochastic functional differential equations with finite delays, Stochastics 6 (2019), 857-872.
  12. X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2007.
  13. B. Oksendal, Stochastic differential Equations: An introduction with Applications, Springer Science and Business Media, 2013.
  14. A.M. Samoilenko, and N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
  15. A. Vinodkumar, M. Gowrisankar and P. Mohankumar, Existence, uniqueness and stability of random impulsive neutral partial differential equations, Journal of the Egyptian Mathematical Society 23 (2015), 31-36. https://doi.org/10.1016/j.joems.2014.01.005
  16. T. Wang and S. Wu, Random impulsive model for stock prices and its application for insurers, Master thesis (in Chinese), Shanghai, East China Normal University, 2008.
  17. S.J. Wu and X.Z. Meng, Boundedness of nonlinear differential systems with impulsive effects on random moments, Acta Mathematicae Applicatae Sinica 20 (2004), 147-154.
  18. X. Yang and Q. Zhu, pth moment exponential stability of stochastic partial differential equations with Poisson jumps, Asian J. Control 6 (2014), 1482-1491.