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On near-optimal necessary and sufficient conditions for forward-backward stochastic systems with jumps, with applications to finance

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Abstract

We establish necessary and sufficient conditions of near-optimality for nonlinear systems governed by forward-backward stochastic differential equations with controlled jump processes (FBSDEJs in short). The set of controls under consideration is necessarily convex. The proof of our result is based on Ekeland’s variational principle and continuity in some sense of the state and adjoint processes with respect to the control variable. We prove that under an additional hypothesis, the near-maximum condition on the Hamiltonian function is a sufficient condition for near-optimality. At the end, as an application to finance, mean-variance portfolio selection mixed with a recursive utility optimization problem is given. Mokhtar Hafay

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. R. Bellman: Dynamic Programming. With a new introduction by Stuart Dreyfus. Reprint of the 1957 edition. Princeton Landmarks in Mathematics, Princeton University Press, Princeton, 2010.

    MATH  Google Scholar 

  3. R. Boel, P. Varaiya: Optimal control of jump processes. SIAM J. Control Optim. 15 (1977), 92–119.

    Article  MATH  MathSciNet  Google Scholar 

  4. B. Bouchard, R. Elie: Discrete-time approximation of decoupled forward-backward SDE with jumps. Stochastic Processes Appl. 118 (2008), 53–75.

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Cadenillas: A stochastic maximum principle for systems with jumps, with applications to finance. Syst. Control Lett. 47 (2002), 433–444.

    Article  MATH  MathSciNet  Google Scholar 

  6. I. Ekeland: On the variational principle. J. Math. Anal. Appl. 47 (1974), 324–353.

    Article  MATH  MathSciNet  Google Scholar 

  7. N. El Karoui, S.G. Peng, M.C. Quenez: Backward stochastic differential equations in finance. Math. Finance 7 (1997), 1–71.

    Article  MATH  MathSciNet  Google Scholar 

  8. N. El Karoui, S.G. Peng, M.C. Quenez: A dynamic maximum principle for the optimization of recursive utilities under constraints. Ann. Appl. Probab. 11 (2001), 664–693.

    Article  MATH  MathSciNet  Google Scholar 

  9. N.C. Framstad, B. Øksendal, A. Sulem: Sufficient stochastic maximum principle for the optimal control of jump diffusions and applications to finance. J. Optim. Theory. Appl. 121 (2004), 77–98; erratum ibid. 124 (2005), 511–512.

    Article  MathSciNet  Google Scholar 

  10. R. Gabasov, F.M. Kirillova, B. Sh. Mordukhovich: The ɛ-maximum principle for suboptimal controls. Sov. Math., Dokl. 27 (1983), 95–99; translation from Dokl. Akad. Nauk SSSR 268 (1983), 525–529. (In Russian.)

    MATH  Google Scholar 

  11. M. Hafayed, S. Abbas, P. Veverka: On necessary and sufficient conditions for near-optimal singular stochastic controls. Optim. Lett. 7 (2013), 949–966.

    Article  MATH  MathSciNet  Google Scholar 

  12. M. Hafayed, P. Veverka, S. Abbas: On maximum principle of near-optimality for diffusions with jumps, with application to consumption-investment problem. Differ. Equ. Dyn. Syst. 20 (2012), 111–125.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Huang, X. Li, G. Wang: Near-optimal control problems for linear forward-backward stochastic systems. Automatica 46 (2010), 397–404.

    Article  MATH  MathSciNet  Google Scholar 

  14. M. Jeanblanc-Picqué, M. Pontier: Optimal portfolio for a small investor in a market model with discontinuous prices. Appl. Math. Optimization 22 (1990), 287–310.

    Article  MATH  Google Scholar 

  15. I. Karatzas, J. P. Lehoczky, S.E. Shreve: Optimal portfolio and consumption decisions for a “Small investor” on a finite horizon. SIAM J. Control Optim. 25 (1987), 1557–1586.

    Article  MATH  MathSciNet  Google Scholar 

  16. B. Sh. Mordukhovich: Approximation Methods in Problems of Optimization and Control. Nauka, Moskva, 1988. (In Russian.)

    MATH  Google Scholar 

  17. B. Øksendal, A. Sulem: Applied Stochastic Control of Jump Diffusions. Second edition. Universitext, Springer, Berlin, 2007.

    Book  Google Scholar 

  18. L.P. Pan, K. L. Teo: Near-optimal controls of class of Volterra integral systems. J. Optimization Theory Appl. 101 (1999), 355–373.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. L. S. Pontryagin, V.G. Boltanskii, R.V. Gamkrelidze, E. F. Mishchenko: The Mathematical Theory of Optimal Processes. Translation from the Russian, Interscience Publishers, New York, 1962.

    Google Scholar 

  21. R. Rishel: A minimum principle for controlled jump processes. Control Theory, Numer. Meth., Computer Syst. Mod.; Internat. Symp. Rocquencourt 1974, Lecture Notes Econ. Math. Syst. 107 (1975), 493–508.

    Article  MathSciNet  Google Scholar 

  22. J. Shi: Necessary conditions for optimal control of forward-backward stochastic systems with random jumps. Int. J. Stoch. Anal. 2012 (2012), Article ID 258674, 50 pp.

  23. J. Shi, Z. Wu: The maximum principle for fully coupled forward-backward stochastic control system. Acta Autom. Sin. 32 (2006), 161–169.

    MathSciNet  Google Scholar 

  24. J. Shi, Z. Wu: Maximum principle for fully coupled forward-backward stochastic control system with random jumps. Proceedings of the 26th Chinese Control Conference, Zhangjiajie, Hunan, 2007, pp. 375–380.

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

    Article  MATH  MathSciNet  Google Scholar 

  26. R. Situ: A maximum principle for optimal controls of stochastic systems with random jumps. Proceedings of National Conference on Control Theory and its Applications. Qingdao, China, 1991.

  27. S. L. Tang, X. J. Li: Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 (1994), 1447–1475.

    Article  MATH  MathSciNet  Google Scholar 

  28. W. Xu: Stochastic maximum principle for optimal control problem of forward and back- ward system. J. Aus. Math. Soc., Ser. B 37 (1995), 172–185.

    Article  MATH  Google Scholar 

  29. J. Yong: Optimality variational principle for controlled forward-backward stochastic differential equations with mixed intial-terminal conditions. SIAM J. Control. Optim. 48 (2010), 4119–4156.

    Article  MATH  MathSciNet  Google Scholar 

  30. J. Yong, X.Y. Zhou: Stochastic Controls. Hamiltonian Systems and HJB Equations. Applications of Mathematics 43, Springer, New York, 1999.

    MATH  Google Scholar 

  31. X.Y. Zhou: Deterministic near-optimal control. I: Necessary and sufficient conditions for near-optimality. J. Optimization Theory Appl. 85 (1995), 473–488.

    Article  MATH  Google Scholar 

  32. X.Y. Zhou: Deterministic near-optimal controls. II: Dynamic programming and viscosity solution approach. Math. Oper. Res. 21 (1996), 655–674.

    Article  MATH  MathSciNet  Google Scholar 

  33. X.Y. Zhou: Stochastic near-optimal controls: Necessary and sufficient conditions for near-optimality. SIAM J. Control. Optim. 36 (1998), 929–947 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  34. X.Y. Zhou, D. Li: Continuous-time mean-variance portfolio selection: A stochastic LQ framework. Appl. Math. Optimization 42 (2000), 19–33.

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratory of Applied Mathematics, Biskra University, P.O. Box 145, 07000, Biskra, Algeria

    Mokhtar Hafayed

  2. Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Trojanova 13, Praha, 120 00, Czech Republic

    Petr Veverka

  3. School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, H.P., 175001, India

    Syed Abbas

Authors
  1. Mokhtar Hafayed
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  2. Petr Veverka
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  3. Syed Abbas
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Corresponding author

Correspondence to Mokhtar Hafayed.

Additional information

The first author was partially supported by Algerian PNR project grant 08/u07/857, ATRST-ANDRU 2011-2013. The second author was supported by the Czech CTU grant SGS12/197/OHK4/3T/14 and MSMT grant INGO II INFRA LG12020.

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Hafayed, M., Veverka, P. & Abbas, S. On near-optimal necessary and sufficient conditions for forward-backward stochastic systems with jumps, with applications to finance. Appl Math 59, 407–440 (2014). https://doi.org/10.1007/s10492-014-0064-4

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  • DOI: https://doi.org/10.1007/s10492-014-0064-4

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