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Stability of Stochastic Functional Differential Equations with Regime-Switching: Analysis Using Dupire’s Functional Itô Formula

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Abstract

This work focuses on almost sure and Lp stability of stochastic functional differential equations by using Lyapunov functionals with the help of the recently developed Dupire’s functional Itô formula. Novel conditions for stability, which are different from those in the existing literature, are given in terms of Lyapunov functionals. It is demonstrated that the conditions are useful for stochastic stabilization. It is also shown that adding a diffusion term can stabilize an unstable system of deterministic differential equations with Markov switching. Furthermore, a robustness result is obtained, which states that the stability of stochastic differential equations with regime-switching is preserved under delayed perturbations when the delay is small enough.

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References

  1. Appleby, J., Mao, X.: Stochastic stabilization of functional differential equations. Syst. Control Lett. 54(11), 1069–1081 (2005)

    Article  MATH  Google Scholar 

  2. Bao, J., Shao, J., Yuan, C.: Approximation of invariant measures for regime-switching diffusions. Potential Anal. 44(4), 707–727 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bao, J., Yin, G., Yuan, C.: Asymptotic analysis for functional stochastic differential equations. Springer, SpringerBriefs in Mathematics (2016)

    Book  MATH  Google Scholar 

  4. Cont, R., Fournié, D.-A.: Change of variable formulas for non-anticipative functionals on path space. J. Funct. Anal. 259(4), 1043–1072 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cont, R., Fournié, D.-A.: Functional Itô calculus and stochastic integral representation of martingales. Ann Probab. 41(1), 109–133 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dupire, B.: Functional Itô’s calculus, bloomberg portfolio research paper no. 2009-04-FRONTIERS available at SSRN: http://ssrn.com/abstract=1435551 or https://doi.org/10.2139/ssrn.1435551

  7. Federico, S., ØKsendal, B.: Optimal stopping of stochastic differential equations with delay driven by Lévy noise. Potential Anal. 34(2), 181–198 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Guo, Q., Mao, X., Yue, R.: Almost sure exponential stability of stochastic differential delay equations. SIAM J. Control Optim. 54(4), 1919–1933 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Khasminskii, R.Z., Zhu, C., Yin, G.: Stability of regime-switching diffusions. Stochastic Process. Appl. 117(8), 1037–1051 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kolmanovskii, V.B., Myshkis, A.: Applied theory of functional differential equations. Kluwer Academic Publishers, Dordrecht (1992)

    Book  MATH  Google Scholar 

  11. Mao, X.: Stochastic functional differential equations with Markovian switching. Funct. Differ. Equ. 6(3-4), 375–396 (1999)

    MathSciNet  MATH  Google Scholar 

  12. Mao, X.: Stability and stabilization of stochastic differential delay equations. IET Control Theory Appl. 1(6), 1551–1566 (2007)

    Article  Google Scholar 

  13. Mao, X.: Stochastic differential equations and their applications, 2nd edn. Horwood, Chichester (2007)

    Google Scholar 

  14. Mao, X., Matasov, A., Piunovskiy, A.: Stochastic differential delay equations with Markovian switching. Bernoulli6 1, 73–90 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mao, X., Yuan, C.: Stochastic differential equations with Markovian Switching. Imperial College Press, London (2006)

    Book  MATH  Google Scholar 

  16. Mohammed, S.-E.A.: Stochastic functional differential equations, Longman Scientific and Technical (1986)

  17. Nguyen, D.H., Yin, G.: Modeling and analysis of switching diffusion systems: past-dependent switching with a countable state space. SIAM J. Control Optim. 54(5), 2450–2477 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nguyen, D.H., Yin, G.: Recurrence and ergodicity of switching diffusions with past-dependent switching having a countable state space. Potential Anal. 48, 405–435 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nguyen, D.H., Yin, G.: Recurrence for switching diffusion with past dependent switching and countable state space. Math. Control Relat. Fields 8(3&4), 879–897 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nguyen, D.H., Yin, G.: Stability of regime-switching diffusion systems with discrete states belonging to a countable set. SIAM J. Control Optim 56, 3893–3917 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pang, T., Hussain, A.: An application of functional Itô’s formula to stochastic portfolio optimization with bounded memory. In: Proceedings of 2015 SIAM conference on control and its applications (CT15), pp. 159–166 (2015)

  22. Scheutzow, M.K.R.: Exponential growth rates for stochastic delay differential equations. Stoch. Dyn. 5(2), 163–174 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shaikhet, L.: Lyapunov functionals and stability of stochastic functional differential equations. Springer, Cham (2013)

    Book  MATH  Google Scholar 

  24. Shaikhet, L.: Stability of stochastic hereditary systems with Markov switching. Theory Stoc. Proc. 2(18), 180–184 (1996)

    MATH  Google Scholar 

  25. Song, M., Mao, X.: Almost sure exponential stability of hybrid stochastic functional differential equations. J. Math. Anal. Appl. 458(2), 1390–1408 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  26. Skorokhod, A.V.: Asymptotic methods in the theory of stochastic differential equations, vol. 78, American Mathematical Soc. (1989)

  27. Tong, J., Jin, X., Zhang, Z.: Exponential Ergodicity for SDEs driven by α-Stable processes with Markovian switching in Wasserstein Distances. Potential Anal. 49(4), 503–526 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yin, G., Zhu, C.: Hybrid switching diffusions: properties and applications. Springer, New York (2010)

    Book  MATH  Google Scholar 

  29. Zhao, X., Deng, F.: New type of stability criteria for stochastic functional differential equations via Lyapunov functions. SIAM J. Control Optim. 52(4), 2319–2347 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zong, X., Yin, G., Wang, L., Li, T., Zhang, J.: Stability of stochastic functional differential systems using degenerate Lyapunov functionals and applications. Automatica J. IFAC 91, 197–207 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Alabama, Tuscaloosa, AL, 35401, USA

    Dang Hai Nguyen

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    George Yin

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  1. Dang Hai Nguyen
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  2. George Yin
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Correspondence to George Yin.

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This research was supported in part by the National Science Foundation under grant DMS-1207667. The research of D. Nguyen was also supported by the AMS-Simons Travel grant.

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Nguyen, D.H., Yin, G. Stability of Stochastic Functional Differential Equations with Regime-Switching: Analysis Using Dupire’s Functional Itô Formula. Potential Anal 53, 247–265 (2020). https://doi.org/10.1007/s11118-019-09767-x

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