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Weak existence and uniqueness for forward-backward SDEs

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  • Delarue, F.
  • Guatteri, G.

Abstract

We aim to establish the existence and uniqueness of weak solutions to a suitable class of non-degenerate deterministic FBSDEs with a one-dimensional backward component. The classical Lipschitz framework is partially weakened: the diffusion matrix and the final condition are assumed to be space Hölder continuous whereas the drift and the backward driver may be discontinuous in x. The growth of the backward driver is allowed to be at most quadratic with respect to the gradient term. The strategy holds in three different steps. We first build a well controlled solution to the associated PDE and as a by-product a weak solution to the forward-backward system. We then adapt the "decoupling strategy" introduced in the four-step scheme of Ma, Protter and Yong [J. Ma, P. Protter, J. Yong, Solving forward-backward stochastic differential equations explicitly -- a four step scheme, Probab. Theory Related Fields 98 (1994) 339-359] to prove uniqueness.

Suggested Citation

  • Delarue, F. & Guatteri, G., 2006. "Weak existence and uniqueness for forward-backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1712-1742, December.
    • Handle: RePEc:eee:spapps:v:116:y:2006:i:12:p:1712-1742
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    References listed on IDEAS

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    1. Delarue, François, 2002. "On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 209-286, June.
    2. Lejay, Antoine, 2004. "A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE," Stochastic Processes and their Applications, Elsevier, vol. 110(1), pages 145-176, March.
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    Cited by:

    1. Martin Herdegen & Johannes Muhle-Karbe & Dylan Possamai, 2019. "Equilibrium Asset Pricing with Transaction Costs," Papers 1901.10989, arXiv.org, revised Sep 2020.
    2. Likibi Pellat, Rhoss & Menoukeu Pamen, Olivier, 2024. "Density analysis for coupled forward–backward SDEs with non-Lipschitz drifts and applications," Stochastic Processes and their Applications, Elsevier, vol. 173(C).
    3. Issoglio, Elena & Jing, Shuai, 2020. "Forward–backward SDEs with distributional coefficients," Stochastic Processes and their Applications, Elsevier, vol. 130(1), pages 47-78.
    4. Bouchemella, Nadira & Raynaud de Fitte, Paul, 2014. "Weak solutions of backward stochastic differential equations with continuous generator," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 927-960.
    5. Gobet, Emmanuel & Makhlouf, Azmi, 2010. "-time regularity of BSDEs with irregular terminal functions," Stochastic Processes and their Applications, Elsevier, vol. 120(7), pages 1105-1132, July.
    6. Luo, Peng & Menoukeu-Pamen, Olivier & Tangpi, Ludovic, 2022. "Strong solutions of forward–backward stochastic differential equations with measurable coefficients," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 1-22.

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