Basic theory and stability analysis for neutral stochastic functional differential equations with pure jumps


This paper investigates the existence and uniqueness of solutions to neutral stochastic functional differential equations with pure jumps (NSFDEwPJs). The boundedness and almost sure exponential stability are also considered. In general, the classical existence and uniqueness theorem of solutions can be obtained under a local Lipschitz condition and linear growth condition. However, there are many equations that do not obey the linear growth condition. Therefore, our first aim is to establish new theorems where the linear growth condition is no longer required whereas the upper bound for the diffusion operator will play a leading role. Moreover, the pth moment boundedness and almost sure exponential stability are also obtained under some loose conditions. Finally, we present two examples to illustrate the effectiveness of our results.

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  1. 1

    Wu L, Zheng W X, Gao H. Dissipativity-based sliding mode control of switched stochastic systems. IEEE Trans Automat Contr, 2013, 58: 785–791

    MathSciNet  Article  Google Scholar 

  2. 2

    Zhang B, Deng F Q, Zhao X Y, et al. Hybrid control of stochastic chaotic system based on memristive Lorenz system with discrete and distributed time-varying delays. IET Contr Theor Appl, 2016, 10: 1513–1523

    MathSciNet  Article  Google Scholar 

  3. 3

    Wu D, Luo X, Zhu S. Stochastic system with coupling between non-Gaussian and Gaussian noise terms. Phys A-Stat Mech Appl, 2007, 373: 203–214

    Article  Google Scholar 

  4. 4

    Shao J, Yuan C. Transportation-cost inequalities for diffusions with jumps and its application to regime-switching processes. J Math Anal Appl, 2015, 425: 632–654

    MathSciNet  Article  Google Scholar 

  5. 5

    Elliott R J, Osakwe C J U. Option pricing for pure jump processes with markov switching compensators. Finance Stochast, 2006, 10: 250–275

    MathSciNet  Article  Google Scholar 

  6. 6

    Lee S S, Mykland P A. Jumps in financial markets: a new nonparametric test and jump dynamics. Rev Financ Stud, 2008, 21: 2535–2563

    Article  Google Scholar 

  7. 7

    Mao W, Zhu Q, Mao X. Existence, uniqueness and almost surely asymptotic estimations of the solutions to neutral stochastic functional differential equations driven by pure jumps. Appl Math Comput, 2015, 254: 252–265

    MathSciNet  MATH  Google Scholar 

  8. 8

    Agarwal R P. Editorial announcement. J Inequal Appl, 2011, 2011: 1

    Article  Google Scholar 

  9. 9

    Song M H, Hu L J, Mao X R, et al. Khasminskii-type theorems for stochastic functional differential equations. Discrete Cont Dyn Syst - Ser B, 2013, 18: 1697–1714

    MathSciNet  MATH  Google Scholar 

  10. 10

    Luo Q, Mao X, Shen Y. Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations. Automatica, 2011, 47: 2075–2081

    MathSciNet  Article  Google Scholar 

  11. 11

    Wu F, Hu S. Khasminskii-type theorems for stochastic functional differential equations with infinite delay. Stat Probab Lett, 2011, 81: 1690–1694

    MathSciNet  Article  Google Scholar 

  12. 12

    Mao X, Rassias M J. Khasminskii-type theorems for stochastic differential delay equations. Stochastic Anal Appl, 2005, 23: 1045–1069

    MathSciNet  Article  Google Scholar 

  13. 13

    Applebaum D. Lévy Processes and Stochastic Calculus. Cambridge: Cambridge University Press, 2009

    Book  Google Scholar 

  14. 14

    Mao X R. Stochastic Differential Equations and Applications. 2nd ed. Cambridge: Woodhead Publishing, 2008

    Book  Google Scholar 

  15. 15

    Meyer P A. Probability and Potentials. Waltham: Blaisdell, 1966

    MATH  Google Scholar 

  16. 16

    Paw lucki W, Plésniak W. Markov’s inequality and C functions on sets with polynomial cusps. Math Ann, 1986, 275: 467–480

    MathSciNet  Article  Google Scholar 

  17. 17

    Beckner W. Inequalities in fourier analysis. Ann Math, 1975, 102: 159–182

    MathSciNet  Article  Google Scholar 

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This work was supported by National Natural Science Foundation of China (Grant Nos. 61573156, 61273126, 61503142), the Ph.D. Start-up Fund of Natural Science Foundation of Guangdong Province (Grant No. 2014A030310388), and Fundamental Research Funds for the Central Universities (Grant No. x2zdD2153620).

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Correspondence to Feiqi Deng.

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Li, M., Deng, F. & Mao, X. Basic theory and stability analysis for neutral stochastic functional differential equations with pure jumps. Sci. China Inf. Sci. 62, 12204 (2019).

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  • neutral term
  • stochastic functional differential equations
  • pure jumps
  • existence and uniqueness theorem
  • stability analysis