logo

SCIENCE CHINA Information Sciences, Volume 62, Issue 1: 012204(2019) https://doi.org/10.1007/s11432-017-9302-9

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

More info
  • ReceivedAug 7, 2017
  • AcceptedNov 6, 2017
  • PublishedOct 16, 2018

Abstract

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 $p$th 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.


Acknowledgment

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).


References

[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 CrossRef Google Scholar

[2] Zhang B, Zhang C, Zhao X. Hybrid control of stochastic chaotic system based on memristive Lorenz system with discrete and distributed time-varying delays. CrossRef Google Scholar

[3] Wu D, Luo X, Zhu S. Stochastic system with coupling between non-Gaussian and Gaussian noise terms. Physica A-Statistical Mech its Appl, 2007, 373: 203-214 CrossRef ADS Google Scholar

[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 CrossRef Google Scholar

[5] Elliott R J, Osakwe C J U. Option Pricing for Pure Jump Processes with Markov Switching Compensators. Finance Stochast, 2006, 10: 250-275 CrossRef Google Scholar

[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 CrossRef Google Scholar

[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 Computation, 2015, 254: 252-265 CrossRef Google Scholar

[8] Agarwal R P. Editorial Announcement. J Inequal Appl, 2011, 2011: 1 CrossRef Google Scholar

[9] Song M, Hu L, Mao X. Khasminskii-type theorems for stochastic functional differential equations. DCDS-B, 2013, 18: 1697-1714 CrossRef Google Scholar

[10] Luo Q, Mao X, Shen Y. Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations. Automatica, 2011, 47: 2075-2081 CrossRef Google Scholar

[11] Wu F, Hu S. Khasminskii-type theorems for stochastic functional differential equations with infinite delay. Stat Probability Lett, 2011, 81: 1690-1694 CrossRef Google Scholar

[12] Mao X, Rassias M J. Khasminskii-Type Theorems for Stochastic Differential Delay Equations. Stochastic Anal Appl, 2005, 23: 1045-1069 CrossRef Google Scholar

[13] Applebaum D. Lévy Processes and Stochastic Calculus. Cambridge: Cambridge University Press, 2009. Google Scholar

[14] Mao X R. Stochastic Differential Equations and Applications. 2nd ed. Cambridge: Woodhead Publishing, 2008. Google Scholar

[15] Meyer P A. Probability and Potentials. Waltham: Blaisdell, 1966. Google Scholar

[16] Paw?ucki W, Ple?niak W. Markov's inequality and $C^\infty$ functions on sets with polynomial cusps. Math Ann, 1986, 275: 467-480 CrossRef Google Scholar

[17] Beckner W. Inequalities in Fourier Analysis. Ann Math, 1975, 102: 159-182 CrossRef Google Scholar

Copyright 2020 Science China Press Co., Ltd. 《中国科学》杂志社有限责任公司 版权所有

京ICP备18024590号-1       京公网安备11010102003388号