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On optimal control of forward–backward stochastic differential equations

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Abstract

We consider a control problem where the system is driven by a decoupled as well as a coupled forward–backward stochastic differential equation. We prove the existence of an optimal control in the class of relaxed controls, which are measure-valued processes, generalizing the usual strict controls. The proof is based on some tightness properties and weak convergence on the space \(\mathcal {D}\) of càdlàg functions, endowed with the Jakubowsky S-topology. Moreover, under some convexity assumptions, we show that the relaxed optimal control is realized by a strict control.

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Acknowledgements

A large part of this work has been carried out when the third author was visiting the Laboratoire LAMAV, Université de Valenciennes (France) in June 2014. He is grateful for warm hospitality and support. The authors would like to thank the anonymous referee for very useful suggestions, which lead to an improvement of the paper.

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Authors and Affiliations

  1. Université de Valenciennes, LAMAV-ISTV2, Mont Houy, 59313, Valenciennes Cedex 9, France

    F. Baghery & I. Turpin

  2. Laboratoire de Mathématiques Appliquées, Université de Biskra, B.P 145, 07000, Biskra, Algeria

    N. Khelfallah & B. Mezerdi

Authors
  1. F. Baghery
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  2. N. Khelfallah
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  3. B. Mezerdi
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  4. I. Turpin
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Corresponding author

Correspondence to B. Mezerdi.

Additional information

Partially supported by French-Algerian Scientific Program PHC Tassili 13 MDU 887.

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Baghery, F., Khelfallah, N., Mezerdi, B. et al. On optimal control of forward–backward stochastic differential equations. Afr. Mat. 28, 1075–1092 (2017). https://doi.org/10.1007/s13370-017-0504-x

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  • DOI: https://doi.org/10.1007/s13370-017-0504-x

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