Exponential stability of uncertain differential equation with jumps

Abstract

Uncertain differential equation with jumps is a type of uncertain differential equation driven by both Liu process and uncertain renewal process. Many concepts of stability for uncertain differential equation with jumps have been investigated. This paper presents a concept of exponential stability for uncertain differential equations with jumps, and gives a sufficient and necessary condition for the linear uncertain differential equation with jumps being exponentially stable. The relationships among stability in measure, stability in in mean, stability in p-th moment, and almost sure stability for uncertain differential equation are also discussed.

Keywords

Uncertain differential equation exponential stability uncertainty theory

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References

Black

and Scholes

, The pricing of options and corporate liabilities, The Journal of Political Economy 81(3) (1973), 637–654.

Chen

, Kar

and Ralescu

D.A.

, Cross-entropy measure of uncertain variables, Information Sciences 201 (2012), 53–60.

Chen

and Ning

, The pth moment exponential stability of uncertain differential equation, Journal of Intelligent & Fuzzy Systems 33 (2017), 725–732.

Jia

, Gen

and Dai

, Uncertain forced vibration equation of spring mass system, Technical Report, 2018.

Gao

and Ahmadzade

, Moment analysis of uncertain stationary independent increment processes, Journal of Uncertain Systems 10(4) (2016), 260–268.

Gao

, Stability in mean for uncertain differential equation with jumps, Applied Mathematics and Computation 346(1) (2019), 15–22.

Tian

, Yang

and Karc

, Equity warrants pricing problem of mean-reverting model in uncertain environment, Physica A: Statistical Mechanics and its Applications, Doi.org/10.1016/j.physa.2019.121593.

Ito

, On stochastic differential equations, Memoirs of the American Mathematical Society 4(8) (1951), 1–51.

and Ke

, Almost sure stability for uncertain differential equation with jumps, Soft Computing 13(4) (2014), 463–473.

10.

Liu

, Uncertainty Theory, Springer-Verlag, Berlin, 2nd edn, 2007.

11.

Liu

, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems 2(1) (2008), 3–16.

12.

Liu

, Some research problems in uncertainty theory, Journal of Uncertain Systems 3(1) (2009), 3–10.

13.

Liu

, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.

14.

Liu

, Why is there a need for uncertainty theory, Journal of Uncertain Systems 6(1) (2012), 3–10.

15.

Liu

, Toward uncertain finance theory, Journal of Uncertainty Analysis and Applications 1 (2013), Article 1.

16.

Liu

, Uncertainty distribution and independence on uncertain processes, Fuzzy Optimization and Decision Making 13(3) (2014), 259–271.

17.

Liu

, Uncertainty Theory, 4th edn, Springer-Verlag, Berlin, 2015.

18.

Liu

, Ke

and Fei

, Almost sure stability for uncertain differential equation, Fuzzy Optimization nd Decision Making 13 (2014), 463–473.

19.

Liu

, Zhang

and Ma

, Uncertain programming models for fixed charge multi-item solid transportation problem, Soft Computing 22(17) (2018), 5825–5833.

20.

, Sheng

, Teng

and Miao

, An uncertain differential equation for SIS epidemic model, Journal of Intelligent & Fuzzy Systems 33(4) (2017), 2317–2327.

21.

, Liu

and Zhang

, Stability in p-th moment for uncertain differential equation with jumps, Journal of Intelligent & Fuzzy Systems 33 (2017), 1375–1384.

22.

, Liu

, Gao

, Zhang

and Zhang

, Stability in distribution for multifactor uncertain differential equation, Journal of Ambient Intelligence and Humanized Computing 8 (2017), 707–716.

23.

Sheng

and Wang

, Stability in p-th moment for uncertain differential equation, Journal of Intelligent and Fuzzy Systems 26(3) (2014), 1263–1271.

24.

Sheng

and Samarjit

, Some results of moments of uncertain variable through inverse uncertainty distribution, Fuzzy Optimization and Decision Making 14(1) (2015), 57–76.

25.

Sheng

and Gao

, Exponential stability of uncertain differential equation, Soft Computing 20 (2016), 3673–3678.

26.

Yao

, Uncertain calulus with renewal process, Fuzzy Optimization and Decision Making 11(3) (2012), 285–297.

27.

Yao

, Uncertain differential equation with jumps, Soft Computing 19 (2015), 2063–2069.

28.

Yao

, Ke

and Sheng

, Stability in mean for uncertain differential equation, Fuzzy Optimization and Decision Making 14(3) (2015), 365–379.

29.

Yang

and Gao

, Uncertain differential games with application to capitalism, Journal of Uncertainty Analysis and Applications 1(17) (2013).

30.

Yang

, Gao

and Kar

Samarjit

, Uncertain calculus with Yao process, IEEE Transactions on Fuzzy Systems 24(6) (2016), 1578–1585.

31.

Yang

, Ni

and Zhang

, Stability in inverse distribution for uncertain differential equations, Journal of Intelligent & Fuzzy Systems 32 (2017), 2051–2059.

32.

Yang

and Yao

, Uncertain partial differential equation with application to heat conduction, Fuzzy Optimization and Decision Making 16(3) (2017), 379–403.

33.

Zhang

, Gao

and Yang

, The stability of multifactor uncertain differential equation, Journal of Intelligent and Fuzzy Systems 30(6) (2016), 3281–3290.

34.

Zhang

and Yang

, Uncertain population model, Soft Computing, to be published.

35.

Zhu

, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems 41(7) (2010), 535–547.