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

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

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.

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/j.spa.2012.08.002
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    References listed on IDEAS

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    1. Wu, Zhen & Xu, Mingyu, 2009. "Comparison theorems for forward backward SDEs," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 426-435, February.
    2. 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.
    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. 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.
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