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On non-Markovian forward–backward SDEs and backward stochastic PDEs


  • Ma, Jin
  • Yin, Hong
  • Zhang, Jianfeng


In this paper, we establish an equivalence relationship between the wellposedness of forward–backward SDEs (FBSDEs) with random coefficients and that of backward stochastic PDEs (BSPDEs). Using the notion of the “decoupling random field”, originally observed in the well-known Four Step Scheme (Ma et al., 1994 [13]) and recently elaborated by Ma et al. (2010) [14], we show that, under certain conditions, the FBSDE is wellposed if and only if this random field is a Sobolev solution to a degenerate quasilinear BSPDE, extending the existing non-linear Feynman–Kac formula to the random coefficient case. Some further properties of the BSPDEs, such as comparison theorem and stability, will also be discussed.

Suggested Citation

  • Ma, Jin & Yin, Hong & Zhang, Jianfeng, 2012. "On non-Markovian forward–backward SDEs and backward stochastic PDEs," Stochastic Processes and their Applications, Elsevier, vol. 122(12), pages 3980-4004.
  • Handle: RePEc:eee:spapps:v:122:y:2012:i:12:p:3980-4004
    DOI: 10.1016/

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    References listed on IDEAS

    1. Ma, Jin & Yong, Jiongmin, 1997. "Adapted solution of a degenerate backward spde, with applications," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 59-84, October.
    2. Hu, Ying & Ma, JinJin, 2004. "Nonlinear Feynman-Kac formula and discrete-functional-type BSDEs with continuous coefficients," Stochastic Processes and their Applications, Elsevier, vol. 112(1), pages 23-51, July.
    3. Wu, Zhen & Xu, Mingyu, 2009. "Comparison theorems for forward backward SDEs," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 426-435, February.
    4. 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.
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    Cited by:

    1. Bernt {O}ksendal & Agn`es Sulem, 2015. "Optimal control of predictive mean-field equations and applications to finance," Papers 1505.04921,
    2. Ulrich Horst & Jinniao Qiu & Qi Zhang, 2014. "A Constrained Control Problem with Degenerate Coefficients and Degenerate Backward SPDEs with Singular Terminal Condition," Papers 1407.0108,, revised Jul 2015.
    3. repec:eee:spapps:v:127:y:2017:i:6:p:1926-1959 is not listed on IDEAS
    4. Stefan Ankirchner & Alexander Fromm & Thomas Kruse & Alexandre Popier, 2017. "Optimal position targeting via decoupling fields," Working Papers hal-01500311, HAL.
    5. Holger Kraft & Thomas Seiferling & Frank Thomas Seifried, 2017. "Optimal consumption and investment with Epstein–Zin recursive utility," Finance and Stochastics, Springer, vol. 21(1), pages 187-226, January.


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