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Fundamental Properties of Nonlinear Stochastic Differential Equations

Author

Listed:
  • Linna Liu

    (School of Electric and Information Engineering, Zhongyuan University of Technology, Zhengzhou 450007, China)

  • Feiqi Deng

    (School of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China)

  • Boyang Qu

    (School of Electric and Information Engineering, Zhongyuan University of Technology, Zhengzhou 450007, China)

  • Yanhong Meng

    (School of Electric and Information Engineering, Zhongyuan University of Technology, Zhengzhou 450007, China)

Abstract

The existence of solutions is used the premise of discussing other properties of dynamic systems. The goal of this paper is to investigate the fundamental properties of nonlinear stochastic differential equations via the Khasminskii test, including the local existence and global existence of the solutions. Firstly, a fundamental result is given as a lemma to verify the local existence of solutions to the considered equation. Then, the equivalent proposition for the global existence and the fundamental principle for the Khasminskii test are formally established. Moreover, the classical Khasminskii test is generalized to the cases with high-order estimates and heavy nonlinearity for the stochastic derivatives of the Lyapunov functions. The role of the noise in this aspect is especially investigated, some concrete criteria are obtained, and an application for the role of the noise in the persistence of financial systems is accordingly provided. As another application of the fundamental principle, a new version of the Khasminskii test is established for the delayed stochastic systems. Finally the conclusions obtained in the paper are verified by simulation. The results show that, under weaker conditions, the global existence of better solutions to stochastic systems to those in the existing literature can be obtained.

Suggested Citation

  • Linna Liu & Feiqi Deng & Boyang Qu & Yanhong Meng, 2022. "Fundamental Properties of Nonlinear Stochastic Differential Equations," Mathematics, MDPI, vol. 10(15), pages 1-18, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2690-:d:875801
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    References listed on IDEAS

    as
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    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    3. Ahmadova, Arzu & Mahmudov, Nazim I., 2020. "Existence and uniqueness results for a class of fractional stochastic neutral differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
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    5. Gao, Shang & Peng, Keyu & Zhang, Chunrui, 2021. "Existence and global exponential stability of periodic solutions for feedback control complex dynamical networks with time-varying delays," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    6. Nane, Erkan & Nwaeze, Eze R. & Omaba, McSylvester Ejighikeme, 2020. "Asymptotic behaviour of solution and non-existence of global solution to a class of conformable time-fractional stochastic equation," Statistics & Probability Letters, Elsevier, vol. 163(C).
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    Cited by:

    1. Yunfeng Li & Pei Cheng & Zheng Wu, 2022. "Exponential Stability of Impulsive Neutral Stochastic Functional Differential Equations," Mathematics, MDPI, vol. 10(21), pages 1-17, November.

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