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Sufficient Conditions of Optimality for Forward-Backward Doubly SDEs with Jumps

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Statistical Methods and Applications in Insurance and Finance (CIMPA School 2013)

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Abstract

We consider a sufficient maximum principle of optimal control for a stochastic control problem. This problem is governed by a system of fully coupled multi-dimensional forward-backward doubly stochastic differential equation with Poisson jumps. Moreover, all the coefficients appearing in this system are allowed to be random and depend on the control variable. We derive, in particular, sufficient conditions for optimality for this stochastic optimal control problem. We apply our result to treat a kind of forward-backward doubly stochastic linear quadratic optimal control problems with jumps.

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References

  1. Ahmed, N.U., Charalambous, C.D.: Stochastic minimum principle for partially observed systems subject to continuous and jump diffusion processes and driven by relaxed controls, arXiv:1302.3455v1 [math.OC], (2013)

  2. Antonelli, F.: Backward-forward stochastic differential equations. Ann. Appl. Probab. 3, 777–793 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bahlali, K., Khelfallah, N., Mezerdi, B.: Necessary and sufficient conditions for near-optimality in stochastic control of FBSDEs. Syst. Control Lett. 58(12), 857–864 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Meng, Q. Sun, Y.: Optimal control problem of fully coupled forward-backward stochastic systems with Poisson jumps under partial information. In: Proceedings at 30th Chinese Control Conference, Yantai, China, 22–24 July 2011, pp. 1317–1322

    Google Scholar 

  5. Nualart, D., Pardoux, E.: Stochastic calculus associated with Skokrohod’s integral, Lecture Notes in Control and Inf. Sciences 96, 363–372 (1987)

    Google Scholar 

  6. Øksendal, B., Sulem, A.: Maximum principles for optimal control of forward-backward stochastic differential equations with jumps. SIAM J. Control Optim. 48(5), 2945–2976 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Pardoux, E., Peng, S.: Backward doubly stochastic differential equations and system of quasilinear SPDEs. Prob. Th. & Rel. Fields 98(2), 209–227 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Peng, S., Shi, Y.: A type of time-symmetric forward-backward stochastic differential equations, C. R. Acad. Sci. Paris Ser. I 336(1), 773–778 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Peng, S., Wu, Z.: Fully coupled forward-backward stochastic differential equations and applications to optimal control. SIAM J. Control Optim 37, 825–843 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Shi, J.: Necessary conditions for optimal control of forward-backward stochastic systems with random jumps, International Journal of Stochastic Analysis, 2012, Article ID 258674, (2012) doi:10.1155/2012/258674

    Google Scholar 

  11. Shi, J., Wu, Z.: The maximum principle for fully coupled forward-backward stochastic control system. Acta Automatica Sinica 32(2), 161–169 (2006)

    MathSciNet  Google Scholar 

  12. Shi, J., Wu, Z.: Maximum principle for partially-observed optimal control of fully-coupled forward-backward stochastic systems. J. Optim. Theory. Appl. 145, 543–578 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Shi, J., Wu, Z.: Maximum principle for forward-backward stochastic control system with random jumps and applications to finance. J. Syst. Sci. Complex. 23(2), 219–231 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Situ, R.: On solution of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl., 66, 209–236 (1997)

    Google Scholar 

  15. Sun, X., Lu, Y.: The property for solutions of the multi-dimensional backward doubly stochastic differential equations with jumps. Chin. J. Appl. Probab. Stat. 24, 73–82 (2008)

    MATH  Google Scholar 

  16. Wu, Z.: Maximum principle for optimal control problem of fully coupled forward-backward stochastic systems. Syst. Sci. Math. Sci. 11(3), 249–259 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Wu, Z.: Forward-backward stochastic differential equations with Brownian motion and Poisson process. Acta Math. Appl. Sinica 15, 433–443 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wu, Z., Wang, X.: FBSDE with Poisson process and its application to linear quadratic stochastic optimal control problem with random jumps. Acta Automatica Sinica 29, 821–826 (2003)

    MathSciNet  Google Scholar 

  19. Xu, S., Jiang, J.: Maximum principle for forward-backward doubly stochastic differential equations with jumps, [OL], http://www.paper.edu.cn/en_releasepaper/content/4486654

  20. Xu, W.: Stochastic maximum principle for optimal control problem of forward and backward system. J. Aust. Math. Soc. B 37, 172–185 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yin, J., Situ, R.: On solutions of forward-backward stochastic differential equations with Poisson jumps. Stoch. Anal. Appl. 21, 1419–1448 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yong, J.: Finding adapted solutions of forward-backward stochastic differential equations-method of continuation. Prob. Th. & Rel. Fields 107, 537–572 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, L., Shi, Y.: General doubly stochastic maximum principle and its applications to optimal control of stochastic partial differential equations, arXiv:1009.6061v3 [math.OC], (2012)

  24. Zhu, Q., Shi, Y.: Forward-backward doubly stochastic differential equations with random jumps and stochastic partial differential-integral equations, [OL].[201001-1044], http://www.paper.edu.cn/index.php/default/en-releasepaper/downPaper/201001-1044

  25. Zhu, Q., Shi, Y.: Forward-backward doubly stochastic differential equations and related stochastic partial differential equations. Sci. Chin. Math. 55(12), 2517–2534 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work is supported by the Science College Research Center at Qassim University, project no. SR-D-012-1958. It is also supported by the Algerian CNEPRU project no. B01420130137.

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Authors and Affiliations

  1. Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah, 51452, Saudi Arabia

    AbdulRahman Al-Hussein

  2. Laboratory of Applied Mathematics, University of Mohamed Khider, P.O. Box 145, Biskra, 07000, Algeria

    Boulakhras Gherbal

Authors
  1. AbdulRahman Al-Hussein
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  2. Boulakhras Gherbal
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Correspondence to AbdulRahman Al-Hussein .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Université Cadi Ayyad Department of Mathematics, Marrakesh, Morocco

    M'hamed Eddahbi

  2. Faculté Polydisciplinaire, Université Cadi Ayyad, Safi, Morocco

    El Hassan Essaky

  3. Faculty of Mathematics, University of Barcelona, Barcelona, Spain

    Josep Vives

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Al-Hussein, A., Gherbal, B. (2016). Sufficient Conditions of Optimality for Forward-Backward Doubly SDEs with Jumps. In: Eddahbi, M., Essaky, E., Vives, J. (eds) Statistical Methods and Applications in Insurance and Finance. CIMPA School 2013. Springer Proceedings in Mathematics & Statistics, vol 158. Springer, Cham. https://doi.org/10.1007/978-3-319-30417-5_7

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