Stochastic maximum principle for partially observed forward-backward stochastic differential equations with jumps and regime switching

Abstract

In this article, we consider the partially observed optimal control problem for forward-backward stochastic systems with Markovian regime switching. A stochastic maximum principle for optimal control is developed using a variational method and filtering technique. Our theoretical results are applied to the motivating example of the risk minimization for portfolio selection.

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References

  1. 1

    Donnelly C. Sufficient stochastic maximum principle in a regime-switching diffusion model. Appl Math Optim, 2011, 64: 155–169

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

    Donnelly C, Heunis A J. Quadratic risk minimization in a regime-switching model with portfolio constraints. SIAM J Control Optim, 2012, 50: 2431–2461

    MathSciNet  Article  MATH  Google Scholar 

  3. 3

    Zhang X, Elliott R J, Siu T K. A stochastic maximum principle for a markov regime-switching jump-diffusion model and its application to finance. SIAM J Control Optim, 2012, 50: 964–990

    MathSciNet  Article  MATH  Google Scholar 

  4. 4

    Zhang Q. Controlled partially observed diffusions with correlated noise. Appl Math Optim, 1990, 22: 265–285

    MathSciNet  Article  MATH  Google Scholar 

  5. 5

    Xiong J, Zhou X Y. Mean-variance portfolio selection under partial information. SIAM J Control Optim, 2007, 46: 156–175

    MathSciNet  Article  MATH  Google Scholar 

  6. 6

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

    MathSciNet  Article  MATH  Google Scholar 

  7. 7

    Wang G C, Wu Z. The maximum principles for stochastic recursive optimal control problems under partial information. IEEE Trans Automat Contr, 2009, 54: 1230–1242

    MathSciNet  Article  MATH  Google Scholar 

  8. 8

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

    MathSciNet  Article  MATH  Google Scholar 

  9. 9

    Øksendal B, Sulem A. Maximum principles for optimal control of forward-backward stochastic differential equations with jumps. SIAM J Control Optim, 2009, 48: 2945–2976

    MathSciNet  Article  MATH  Google Scholar 

  10. 10

    Huang J, Wang G, Xiong J. A Maximum principle for partial information backward stochastic control problems with applications. SIAM J Control Optim, 2009, 48: 2106–2117

    MathSciNet  Article  MATH  Google Scholar 

  11. 11

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

    MathSciNet  Article  MATH  Google Scholar 

  12. 12

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

    MathSciNet  Article  MATH  Google Scholar 

  13. 13

    Wang G C, Wu Z. Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems. J Math Anal Appl, 2008, 342: 1280–1296

    MathSciNet  Article  MATH  Google Scholar 

  14. 14

    Wang G C, Wu Z. General maximum principles for partially observed risk-sensitive optimal control problems and applications to finance. J Optim Theor Appl, 2009, 141: 677–700

    Article  MATH  Google Scholar 

  15. 15

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

    MathSciNet  Article  MATH  Google Scholar 

  16. 16

    Wang G C, Wu Z, Xiong J. An introduction to optimal control of FBSDE with incomplete information. In: Springer- Briefs in Mathematics. Berlin: Springer, 2018

    Book  MATH  Google Scholar 

  17. 17

    Huang J, Zhang D. The near-optimal maximum principle of impulse control for stochastic recursive system. Sci China Inf Sci, 2016, 59: 112206

    Article  Google Scholar 

  18. 18

    Elliott R J, Aggoun L, Moore J B. Hidden Markov Models: Estimation and Control. New York: Springer, 1994. 176–182

    Google Scholar 

  19. 19

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

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

ZHANG’s research was supported in part by National Natural Science Foundation of China (Grant Nos. 11501129, 71571053, 71771058) and Natural Science Foundation of Hebei Province (Grant No. A2014202202). XIONG’s research was supported by Macao Science and Technology Fund FDCT (Grant No. FDCT025/2016/A1).

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Correspondence to Shuaiqi Zhang.

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Zhang, S., Xiong, J. & Liu, X. Stochastic maximum principle for partially observed forward-backward stochastic differential equations with jumps and regime switching. Sci. China Inf. Sci. 61, 70211 (2018). https://doi.org/10.1007/s11432-017-9267-0

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Keywords

  • partial information
  • Markovian regime-switching
  • stochastic maximum principle
  • forward-backward stochastic differential equation (FBSDE)