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Mean-Field-Type Games with Jump and Regime Switching

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Abstract

In this article, we study mean-field-type games with jump–diffusion and regime switching in which the payoffs and the state dynamics depend not only on the state–action profile of the decision-makers but also on a measure of the state–action pair. The state dynamics is a measure-dependent process with jump–diffusion and regime switching. We derive novel equilibrium systems to be solved. Two solution approaches are presented: (i) dynamic programming principle and (ii) stochastic maximum principle. Relationship between dual function and adjoint processes are provided. It is shown that the extension to the risk-sensitive case generates a nonlinearity to the adjoint process and it involves three other processes associated with the diffusion, jump and regime switching, respectively.

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References

  1. 1.

    Agram N Bernt O (2017) Stochastic control of memory mean-field processes. arXiv:1701.01801

  2. 2.

    Agram N, Hu Y, Oksendal B (2018) Mean-field backward stochastic differential equations and applications. arXiv:1801.03349v2

  3. 3.

    Andersson D, Djehiche B (2011) A maximum principle for stochastic control of SDE’s of mean-field type. Appl Math Optim 63(3):341–356

  4. 4.

    Augustin F, Gilg A, Paffrath M, Rentrop P, Wever U (2008) Polynomial chaos for the approximation of uncertainties: chances and limits. Eur J Appl Math 19:149–190. https://doi.org/10.1017/S0956792508007328

  5. 5.

    Bayraktar E, Cosso A, Pham H (2016) Randomized dynamic programming principle and Feynman–Kac representation for optimal control of McKean–Vlasov dynamics. arXiv:1606.08204

  6. 6.

    Bensoussan A, Frehse J, Yam SCP (2013) Mean-field games and mean-field-type control theory. Springer briefs in mathematics. Springer, Berlin

  7. 7.

    Bensoussan A, Frehse J, Yam SCP (2017) On the interpretation of the master equation. Stoch Process Appl 127(7):2093–2137

  8. 8.

    Bensoussan A, Djehiche B, Tembine H, Yam P (2017) Risk-sensitive mean-field-type control. In: IEEE 56th annual conference on decision and control (CDC), 12–15 Dec. Melbourne, Australia

  9. 9.

    Bensoussan A, Yam P (2018) Control problem on space of random variables and master equation, ESAIM: COCV, Arxiv: 1508.00713

  10. 10.

    Buckdahn R, Djehiche B, Li J, Peng S (2009) Mean-field backward stochastic differential equations: a limit approach. Ann Probab 37:1524–1565

  11. 11.

    Buckdahn R, Li J, Peng S (2009) Mean-field backward stochastic differential equations and related partial differential equations. Stoch Process Appl 119:3133–3154

  12. 12.

    Buckdahn R, Djehiche B, Li J (2011) A general stochastic maximum principle for SDEs of mean-field type. Appl Math Optim 64(2):197–216

  13. 13.

    Cameron RH, Martin WT (1947) The orthogonal development of non-linear functionals in series of Fourier–Hermite functionals. Ann Math 48:385–392

  14. 14.

    Carmona R, Delarue F (2014) The master equation for large population equilibriums. In: Crisan D, Hambly B, Zariphopoulou T (eds) Stochastic analysis and applications. Springer, Berlin

  15. 15.

    Carmona R, Delarue F (2014) Forward-backward stochastic differential equations and controlled McKean Vlasov dynamics. Ann Probab 43:2647–2700

  16. 16.

    Carmona R, Delarue F, Lachapelle A (2013) Control of McKean–Vlasov dynamics versus mean-field games. Math Finance Econ 7:131–166

  17. 17.

    Cisse AK, Tembine H (2014) Cooperative mean-field type games. In: 19th world congress of the international federation of automatic control (IFAC), pp 8995–9000, Cape Town, South Africa, 24–29 Aug 2014

  18. 18.

    Djehiche B, Tembine H (2016) Risk-sensitive mean-field-type control under partial observation. In: Benth FE, Di Nunno G (eds) Stochastics in environmental and financial economics. Proceedings in mathematics and statistics. Springer, Berlin

  19. 19.

    Djehiche B, Tembine H, Tempone R (2015) A stochastic maximum principle for risk-sensitive mean-field-type control. IEEE Trans Autom Control 60(10):2640–2649

  20. 20.

    Djehiche B, Tcheukam A, Tembine H (2017) Mean-field-type games in engineering. AIMS Electron Electr Eng 1(4):18–73

  21. 21.

    Djehiche B, Tcheukam SA, Tembine H (2017) A mean-field game of evacuation in multi-level building. IEEE Trans Autom Control 62(10):5154–5169

  22. 22.

    Elliott R, Li X, Ni YH (2013) Discrete time mean-field stochastic linear-quadratic optimal control problems. Automatica 49(11):3222–3233

  23. 23.

    Gao J, Tembine H (2016) Distributed mean-field-type filters for big data assimilation. In: IEEE international conference on data science systems (DSS 2016), Sydney, Australia 12–14 Dec

  24. 24.

    Hafayed M (2013) A mean-field maximum principle for optimal control of forward-backward stochastic differential equations with Poisson jump processes. Int J Dyn Control 1(4):300–315

  25. 25.

    Hafayed M, Abbas S, Abba A (2015) On mean-field partial information maximum principle of optimal control for stochastic systems with Lévy processes. J Optim Theory Appl 167(3):1051–1069

  26. 26.

    Hosking J (2012) A stochastic maximum principle for a stochastic differential game of a mean-field type. Appl Math Optim 66:415–454

  27. 27.

    Jourdain B, Méléard S, Woyczynski W (2008) Nonlinear SDEs driven by Lévy processes and related PDEs. Alea 4:1–29

  28. 28.

    Jovanovic B, Rosenthal RW (1988) Anonymous sequential games. J Math Econ 17(1):77–87

  29. 29.

    Kolokoltsov VN (2012) Nonlinear Markov games on a finite state space (mean-field and binary interactions) arXiv:1105.3053. Int J Stat Prob. Canadian Center of Science and Education (Open access journal), 1:1, 77–91. http://www.ccsenet.org/journal/index.php/ijsp/article/view/16682

  30. 30.

    Kolokoltsov V, Troeva M, Yang W (2014) On the rate of convergence for the mean-field approximation of controlled diffusions with large number of players. Dyn Games Appl 4(2):208–230

  31. 31.

    Law K, Tembine H, Tempone R (2016) Deterministic mean-field ensemble Kalman filtering. SIAM J Sci Comput (SISC) 38(3):A1215–1279

  32. 32.

    Li J (2012) Stochastic maximum principle in the mean-field controls. Automatica 48:366–373

  33. 33.

    Ma H, Liu B (2017) Linear-quadratic optimal control problem for partially observed forward backward stochastic differential equations of mean-field type. Asian J Control 18:2146–2157

  34. 34.

    Meng Q, Shen Y (2015) Optimal control of mean-field jump–diffusion systems with delay: a stochastic maximum principle approach. J Comput Appl Math 279:13–30

  35. 35.

    Meyer-Brandis T, Oksendal B, Zhou XY (2012) A mean-field stochastic maximum principle via Malliavin calculus. Stochastics 84:643–666

  36. 36.

    Petrosjan LA (1977) Stability of the solutions in differential games with several players. Russian vestnik leningrad. Univ. no. 19 Mat. Mat. Meh. Astronom. 4, 4652,147

  37. 37.

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

  38. 38.

    Rogers LCG, Williams D (2000) Diffusions, Markov processes and martingales-volume 2: Itô calculus. Cambridge University Press, Cambridge

  39. 39.

    Rossi G, Tcheukam AS, Tembine H (2016) How much does users’ psychology matter in engineering mean-field-type games. In: Workshop on game theory and experimental methods, 6–7 June. Second University of Naples, Italy

  40. 40.

    Shapley LS (1953) Stochastic games. PNAS Proc Natl Acad Sci USA 39(10):1095–1100

  41. 41.

    Shen Y, Siu TK (2013) The maximum principle for a jump–diffusion mean-field model and its application to the mean-variance problem. Nonlinear Anal Theory Methods Appl 86:58–73

  42. 42.

    Shen Y, Meng Q, Shi P (2014) Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance. Automatica 50(6):1565–1579

  43. 43.

    Tang S (1998) The maximum principle for partially observed optimal control of stochastic differential equations. SIAM J Control Optim 36(5):1596–1617

  44. 44.

    Tcheukam AS, Tembine H (2016) On the distributed mean-variance paradigm. In: 13th international multi-conference on systems, signals & devices. Conference on systems, automation & control, 21–24 March 2016. Leipzig, Germany, pp 604–609

  45. 45.

    Tembine H (2014) Tutorial on mean-field-type games. In: 19th world congress of the international federation of automatic control (IFAC), Cape Town, South Africa, pp 24–29

  46. 46.

    Tembine H (2015) Uncertainty quantification in mean-field-type teams and games. In: Proceedings of 54th IEEE conference on decision and control (CDC), pp 4418–4423

  47. 47.

    Tembine H (2015) Risk-sensitive mean-field-type games with Lp-norm drifts. Automatica 59:224–237

  48. 48.

    Tembine H (2017) Mean-field-type games. AIMS Math 2(4):706–735

  49. 49.

    Wang G, Wu Z (2009) The maximum principles for stochastic recursive optimal control problems under partial information. IEEE Trans Autom control 54(6):1230–1242

  50. 50.

    Wang G, Wu Z, Xiong J (2013) Maximum principles for forward–backward stochastic control systems with correlated state and observation noises. SIAM J Control Optim 51(1):491–524

  51. 51.

    Wang G, Zhang C, Zhang W (2014) Stochastic maximum principle for mean-field type optimal control under partial information. IEEE Trans Autom Control 59(2):522–528

  52. 52.

    Wang G, Wu Z, Xiong J (2015) A linear-quadratic optimal control problem of forward–backward stochastic differential equations with partial information. IEEE Trans Autom Control 60(11):2904–2916

  53. 53.

    Wiener N (1938) The homogeneous chaos. Am J Math 60:897–936

  54. 54.

    Wiener N (1958) Nonlinear problems in random theory. Technology Press of the Massachusetts Institute of Technology/Wiley, New York

  55. 55.

    Wu Z (2010) A maximum principle for partially observed optimal control of forward–backward stochastic control systems. Sci China Inf Sci 53(11):2205–2214

  56. 56.

    Yong J (2013) Linear-quadratic optimal control problems for mean-field stochastic differential equations. SIAM J Control Optim 51(4):2809–2838

  57. 57.

    Zhang X, Elliott RJ, Siu TK (2012) A stochastic maximum principle for a Markov regime-switching jump–diffusion model and its application to finance. SIAM J Control Optim 50:964–990

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Acknowledgements

Alain Bensoussan is supported by Grants from the National Science Foundation NSF-DMS 1612880 and the Research Grants Council of the Hong Kong Special Administrative Region City U-11303316. The research of Boualem Djehiche is supported by Grants from the Swedish Research Council 2016-04086. The research of Hamidou Tembine is supported by US Air Force Office of Scientific Research under Grant Number FA9550-17-1-0259 with title: Mean-Field-Type Games. Sheung Chi Phillip Yam acknowledges the financial support from HKGRF-14300717 with the project title: New Kinds of Forward–Backward Stochastic Systems with Applications, HKSAR-GRF-14301015 with title: Advance in Mean Field Theory, Direct Grant for Research 2014/15 with project code: 4053141 offered by CUHK.

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Correspondence to Hamidou Tembine.

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Bensoussan, A., Djehiche, B., Tembine, H. et al. Mean-Field-Type Games with Jump and Regime Switching. Dyn Games Appl 10, 19–57 (2020). https://doi.org/10.1007/s13235-019-00306-2

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Keywords

  • Mean-field
  • McKean–Vlasov
  • Game theory