On non-Markovian forward–backward SDEs and backward stochastic PDEs
On non-Markovian forward–backward SDEs and backward stochastic PDEs
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DOI：
10.1016/j.spa.2012.08.002
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发表时间：
2012-12
影响因子：
1.4
通讯作者：
Jin Ma;H. Yin;Jianfeng Zhang
中科院分区：
文献类型：
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作者：
Jin Ma;H. Yin;Jianfeng Zhang
本文建立了随机系数正倒向随机微分方程（FBSDEs）与倒向随机偏微分方程（BSPDEs）的适定性之间的等价关系。使用最初在众所周知的四步方案（Ma等人，1994 [13]），最近由Ma et al.（2010）[14]中，我们证明了在一定条件下，FBBDE是适定的当且仅当这个随机场是退化拟线性BSPDE的Sobolev解，将现有的非线性Feynman-Kac公式推广到随机系数的情况。一些进一步的性质，如比较定理和稳定性，也将讨论。
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.