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

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

    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 122 (2012)
    Issue (Month): 12 ()
    Pages: 3980-4004

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    Handle: RePEc:eee:spapps:v:122:y:2012:i:12:p:3980-4004

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    Keywords: Forward–backward stochastic differential equations; Backward stochastic partial differential equations; Nonlinear stochastic Feynman–Kac formula;

    References

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    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. Wu, Zhen & Xu, Mingyu, 2009. "Comparison theorems for forward backward SDEs," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 426-435, February.
    3. 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.
    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. 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, arXiv.org.

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