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Propagation of chaos and conditional McKean-Vlasov SDEs with regime-switching

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Abstract

We investigate a particle system with mean field interaction living in a random environment characterized by a regime-switching process. The switching process is allowed to be dependent on the particle system. The well-posedness and various properties of the limit conditional McKean-Vlasov SDEs are studied, and the conditional propagation of chaos is established with explicit estimate of the convergence rate.

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References

  1. Andreis L, Pra P D, Fischer M. McKean-Vlasov limit for interacting systems with simultaneous jumps. Stoch Anal Appl, 2018, 36: 960–995

    Article  MathSciNet  MATH  Google Scholar 

  2. Budhiraja A, Kira E, Saha S. Central limit results for jump diffusions with mean field interaction and a common factor. Stoch Anal Appl, 2017, 35: 767–802

    Article  MathSciNet  MATH  Google Scholar 

  3. Carmona R, Delarue F. Probabilistic Theory of Mean Field Games with Applications II. Probab Theory Stoch Model, Vol 84. Berlin: Springer, 2018

    Book  MATH  Google Scholar 

  4. Carmona R, Delarue F, Lacker D. Mean field games with common noise. Ann Probab, 2016, 44: 3740–3803

    Article  MathSciNet  MATH  Google Scholar 

  5. Carmona R, Zhu X N. A probabilistic approach to mean field games with major and minor players. Ann Appl Probab, 2016, 26: 1535–1580

    Article  MathSciNet  MATH  Google Scholar 

  6. Djete M F, Possamaï D, Tan X L. McKean-Vlasov optimal control: the dynamic programming principle. arXiv: 1907.08860

  7. Erny X, Löcherbach E, Loukianova D. White-noise driven conditional McKean-Vlasov limits for systems of particles with simultaneous and random jumps. arXiv: 2103.04100

  8. Fournier N, Guillin A. On the rate of convergence in Wasserstein distance of the empirical measure. Probab Theory Related Fields, 2014, 162: 707–738

    Article  MathSciNet  MATH  Google Scholar 

  9. Gärtner, J. On the McKean-Vlasov limit for interacting diffusions. Math Nachr, 1988, 137: 197–248

    Article  MathSciNet  MATH  Google Scholar 

  10. Graham C. McKean-Vlasov Itô-Skorohod equations, and nonlinear diffusions with discrete jump sets. Stochastic Process Appl, 1992, 40: 69–82

    Article  MathSciNet  MATH  Google Scholar 

  11. Graham C, Méléard S. Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann Probab, 1997, 25: 115–132

    Article  MathSciNet  MATH  Google Scholar 

  12. Hammersley W R P, Siska D, Szpruch L. Weak existence and uniqueness for McKean-Vlasov SDEs with common noise. Ann Probab, 2021, 49: 527–555

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang X, Wang F-Y. Distribution dependent SDEs with singular coefficients. Stochastic Process Appl, 2019, 129: 4747–4770

    Article  MathSciNet  MATH  Google Scholar 

  14. Kac M. Foundations of Kinetic Theory. Berkeley: Univ of California Press, 2020, 173–200

    Google Scholar 

  15. McKean J, Henry P. A class of Markov processes associated with nonlinear parabolic equations. Proc Natl Acad Sci USA, 1996, 56: 1907

    Article  MathSciNet  MATH  Google Scholar 

  16. Méléard S. Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models. In: Talay D, Tubaro L, eds. Probabilistic Models for Nonlinear Partial Differential Equations. Lecture Notes in Math, Vol 1627. Berlin: Springer, 1996, 42–95

    Chapter  MATH  Google Scholar 

  17. Nguyen D H, Yin G, Zhu C. Certain properties related to well posedness of switching diffusions. Stochastic Process Appl, 2017, 127: 3135–3158

    Article  MathSciNet  MATH  Google Scholar 

  18. Nguyen S L, Yin G, Hoang T A. On laws of large numbers for systems with mean-field interactions and Markovian switching. Stochastic Process Appl, 2020, 130: 262–296

    Article  MathSciNet  MATH  Google Scholar 

  19. Oelschlager K. A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann Probab, 1984, 458–479

  20. Pham H, Wei X L. Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics. SIAM J Control Optim, 2017, 55: 1069–1101

    Article  MathSciNet  MATH  Google Scholar 

  21. Pinsky M, Pinsky R G. Transience/recurrence and central limit theorem behavior for diffusions in random temporal environments. Ann Probab, 1993, 21: 433–452

    Article  MathSciNet  MATH  Google Scholar 

  22. Shao J H. Criteria for transience and recurrence of regime-switching diffusion processes. Electron J Probab, 2015, 20: 1–15

    Article  MathSciNet  MATH  Google Scholar 

  23. Shao J H. Strong solutions and strong Feller properties for regime-switching diffusion processes in an infinite state space. SIAM J Control Optim, 2015, 53: 2462–2479

    Article  MathSciNet  MATH  Google Scholar 

  24. Shao J H, Zhao K. Continuous dependence for stochastic functional differential equations with state-dependent regime-switching on initial values. Acta Math Sin (Engl Ser), 2021, 37: 389–407

    Article  MathSciNet  MATH  Google Scholar 

  25. Sznitman A S. Topics in propagation of chaos. In: Burkholder D L, Pardoux E, Sznitman A, eds. Ecole d’Eté de Probabilités de Saint-Flour XIX–1989. Lecture Notes in Math, Vol 1464. Berlin: Springer, 1991, 165–251

    Chapter  Google Scholar 

  26. Yin G, Zhu C. Hybrid Switching Diffusions: Properties and Applications. New York: Springer, 2009

    MATH  Google Scholar 

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11771327, 11831014).

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Authors and Affiliations

  1. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Jinghai Shao & Dong Wei

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  1. Jinghai Shao
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  2. Dong Wei
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Correspondence to Dong Wei.

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Shao, J., Wei, D. Propagation of chaos and conditional McKean-Vlasov SDEs with regime-switching. Front. Math. China 17, 731–746 (2022). https://doi.org/10.1007/s11464-021-0960-3

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