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INFINITE HORIZON OPTIMAL CONTROL PROBLEMS OF BACKWARD STOCHASTIC DELAY DIFFERENTIAL EQUATIONS IN HILBERT SPACES

  • Liang, Hong (Institute of Applied Mathematics College of Science Northwest A&F University) ;
  • Zhou, Jianjun (Institute of Applied Mathematics College of Science Northwest A&F University)
  • Received : 2019.02.21
  • Accepted : 2019.05.16
  • Published : 2020.03.31

Abstract

This paper investigates infinite horizon optimal control problems driven by a class of backward stochastic delay differential equations in Hilbert spaces. We first obtain a prior estimate for the solutions of state equations, by which the existence and uniqueness results are proved. Meanwhile, necessary and sufficient conditions for optimal control problems on an infinite horizon are derived by introducing time-advanced stochastic differential equations as adjoint equations. Finally, the theoretical results are applied to a linear-quadratic control problem.

Acknowledgement

Supported by : Natural Science Foundation of Shaanxi Province, Central Universities

References

  1. A. Bensoussan, Lectures on stochastic control, in Nonlinear filtering and stochastic control (Cortona, 1981), 1-62, Lecture Notes in Math., 972, Springer, Berlin, 1982.
  2. L. Chen and J. Huang, Stochastic maximum principle for controlled backward delayed system via advanced stochastic differential equation, J. Optim. Theory Appl. 167 (2015), no. 3, 1112-1135. https://doi.org/10.1007/s10957-013-0386-5
  3. J. Cvitanic, I. Karatzas, and H. M. Soner, Backward stochastic differential equations with constraints on the gains-process, Ann. Probab. 26 (1998), no. 4, 1522-1551. https://doi.org/10.1214/aop/1022855872
  4. L. Delong and P. Imkeller, Backward stochastic differential equations with time delayedgenerators|results and counterexamples, Ann. Appl. Probab. 20 (2010), no. 4, 1512-1536. https://doi.org/10.1214/09-AAP663
  5. M. Fuhrman and G. Tessitore, Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces, Ann. Probab. 32 (2004), no. 1B, 607-660. https://doi.org/10.1214/aop/1079021459
  6. Y. Hu and S. G. Peng, Maximum principle for semilinear stochastic evolution control systems, Stochastics Stochastics Rep. 33 (1990), no. 3-4, 159-180. https://doi.org/10.1080/17442509008833671
  7. N. El Karoui, S. Peng, and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance 7 (1997), no. 1, 1-71. https://doi.org/10.1111/1467-9965.00022
  8. V. B. Kolmanovskii and T. L. Maizenberg, Optimal control of stochastic systems with after effect, Automat. Remote Control 34 (1973), no. 1, part 1, 39-52; translated from Avtomat. i Telemeh. 1973, no. 1, 47-61.
  9. J. Li and S. Peng, Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton-Jacobi-Bellman equations, Nonlinear Anal. 70 (2009), no. 4, 1776-1796. https://doi.org/10.1016/j.na.2008.02.080
  10. X. Mao, Adapted solutions of backward stochastic differential equations with non-Lipschitz coecients, Stochastic Process. Appl. 58 (1995), no. 2, 281-292.
  11. Pardoux and S. G. 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
  12. M. Riedle, Cylindrical Wiener processes, arXiv:0802.2261v1 [math.PR], 2008.
  13. S. J. Tang and X. J. Li, Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim. 32 (1994), no. 5, 1447-1475. https://doi.org/10.1137/S0363012992233858
  14. S. Wang and Z. Wu, Stochastic maximum principle for optimal control problems of forward-backward delay systems involving impulse controls, J. Syst. Sci. Complex. 30 (2017), no. 2, 280-306. https://doi.org/10.1007/s11424-016-5039-y
  15. J. Yin and X. Mao, The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications, J. Math. Anal. Appl. 346 (2008), no. 2, 345-358. https://doi.org/10.1016/j.jmaa.2008.05.072
  16. Z. Yu, Infinite horizon jump-diusion forward-backward stochastic differential equations and their application to backward linear-quadratic problems, ESAIM Control Optim. Calc. Var. 23 (2017), no. 4, 1331-1359.