Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

MEAN-FIELD BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS ON MARKOV CHAINS

  • Lu, Wen (School of Mathematics and Informational Science Yantai University) ;
  • Ren, Yong (Department of Mathematics Anhui Normal University)
  • Received : 2015.01.05
  • Published : 2017.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we deal with a class of mean-field backward stochastic differential equations (BSDEs) related to finite state, continuous time Markov chains. We obtain the existence and uniqueness theorem and a comparison theorem for solutions of one-dimensional mean-field BSDEs under Lipschitz condition.

Keywords

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. N. Ahmed, Nonlinear diffusion governed by McKean-Vlasov equation on Hilbert space and optimal control, SIAM J. Control Optim. 46 (2007), no. 1, 356-378. https://doi.org/10.1137/050645944
  2. N. Ahmed and X. Ding, A semilinear McKean-Vlasov stochastic evolution equation in Hilbert space, Stochastic Process. Appl. 60 (1995), no. 1, 65-85. https://doi.org/10.1016/0304-4149(95)00050-X
  3. V. Borkar and K. Kumar, McKean-Vlasov limit in portfolio optimization, Stoch. Anal. Appl. 28 (2010), no. 5, 884-906. https://doi.org/10.1080/07362994.2010.482836
  4. R. Buckdahn, B. Djehiche, J. Li, and S. Peng, Mean-field backward stochastic differential equations: a limit approach, Ann. Probab. 37 (2009), no. 4, 1524-1565. https://doi.org/10.1214/08-AOP442
  5. R. Buckdahn, J. Li, and S. Peng,Mean-field backward stochastic differential equations and related partial differential equations, Stoch. Process. Appl. 119 (2009), no. 10, 3133-3154. https://doi.org/10.1016/j.spa.2009.05.002
  6. T. Chan, Dynamics of the McKean-Vlasov equation, Ann. Probab. 22 (1994), no. 1, 431-441. https://doi.org/10.1214/aop/1176988866
  7. S. N. Cohen and R. J. Elliott, Solutions of backward stochastic differential equations on Markov chains, Commun. Stoch. Anal. 2 (2008), no. 2, 251-262.
  8. S. N. Cohen and R. J. Elliott, Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions, Ann. Appl. Probab. 20 (2010), no. 1, 267-311. https://doi.org/10.1214/09-AAP619
  9. S. N. Cohen and R. J. Elliott, Existence, uniqueness and comparisons for BSDEs in general spaces, Ann. Probab. 40 (2012), no. 5, 2264-2297. https://doi.org/10.1214/11-AOP679
  10. D. Crisan and J. Xiong, Approximate McKean-Vlasov representations for a class of SPDEs, Stochastics 82 (2010), no. 1-3, 53-68. https://doi.org/10.1080/17442500902723575
  11. N. El Karoui, S. Peng, and M. Quenez, Backward stochastic differential equations in finance, Math. Finance 7 (1997), no. 1, 1-71. https://doi.org/10.1111/1467-9965.00022
  12. R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden Markov Models: Estimation and Control, Springer-Verlag, Berlin-Heidelberg-New York, 1994.
  13. S. Hamadene and J. Lepeltier, Zero-sum stochastic differential games and backward equations, Systems Control Lett. 24 (1995), no. 4, 259-263. https://doi.org/10.1016/0167-6911(94)00011-J
  14. M. Kac, Foundations of kinetic theory, In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954955, vol. III, pp. 171-197. University of California Press, Berkeley and Los Angeles, 1956.
  15. P. Kotelenez, A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation, Probab. Theory Related Fields 102 (1995), no. 2, 159-188. https://doi.org/10.1007/BF01213387
  16. P. Kotelenez and T. Kurtz,Macroscopic limit for stochastic partial differential equations of McKean-Vlasov type, Probab. Theory Related Fields 146 (2010), no. 1-2, 189-222. https://doi.org/10.1007/s00440-008-0188-0
  17. J. Lasry and P. Lions, Mean field games, Jpn J. Math. 2 (2007), no. 1, 229-260. https://doi.org/10.1007/s11537-007-0657-8
  18. J. Li, Reflected Mean-Field Backward Stochastic Differential Equations, Approximation and Associated Nonlinear PDEs, J. Math. Anal. Appl. 413 (2014), no. 1, 47-68. https://doi.org/10.1016/j.jmaa.2013.11.028
  19. Z. Li and J. Luo, Mean-field reflected backward stochastic differential equations, Statist. Probab. Lett. 82 (2012), no. 11, 1961-1968. https://doi.org/10.1016/j.spl.2012.06.018
  20. W. Lu and Y. Ren, Anticipated backward stochastic differential equations on Markov chains, Statist. Probab. Lett. 83 (2013), no. 7, 1711-1719. https://doi.org/10.1016/j.spl.2013.03.022
  21. H. McKean, A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. U.S.A. 56 (1966), 1907-1911. https://doi.org/10.1073/pnas.56.6.1907
  22. E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55-61. https://doi.org/10.1016/0167-6911(90)90082-6
  23. S. Peng, Backward stochastic differential equations and applications to optimal control, Appl. Math. Optim. 27 (1993), no. 2, 125-144. https://doi.org/10.1007/BF01195978
  24. R. Xu, Mean-field backward doubly stochastic differential equations and related SPDEs, Bound. Value Probl. 2012 (2012), 114, 20 pp. https://doi.org/10.1186/1687-2770-2012-114

Cited by

  1. Reflection positivity, duality, and spectral theory pp.1865-2085, 2018, https://doi.org/10.1007/s12190-018-1184-x