In this work, we study the numerical approximation of a class of singular fully coupled forward backward stochastic differential equations. These equations have a degenerate forward component and non-smooth terminal condition. They are used, for example, in the modelling of carbon market [1] and are linked to scalar conservation law perturbed by a diffusion. Classical FBSDEs methods fail to capture the correct entropy solution to the associated quasi-linear PDE. We introduce a splitting approach that circumvent this difficulty by treating differently the numerical approximation of the diffusion part and the non-linear transport part. Under the structural condition guaranteeing the well-posedness of the singular FBSDEs [2], we show that the splitting method is convergent with a rate 1/2. We implement the splitting scheme combining non-linear regression based on deep neural networks and conservative finite difference schemes. The numerical tests show very good results in possibly high dimensional framework.

在这项工作中，我们研究了一类奇异的全耦合正反向随机微分方程的数值逼近。这些方程具有退化的前向分量和非光滑终端条件。例如，它们用于碳市场建模 [1]，并与受扩散扰动的标量守恒定律相关联。经典的 FBSDEs 方法无法捕获相关准线性 PDE 的正确熵解。我们引入了一种分裂方法，通过不同地处理扩散部分和非线性传输部分的数值近似来规避这个困难。在保证奇异 FBSDE [2] 适定性的结构条件下，我们证明了分裂方法以 1/2 的速率收敛。我们实现了结合基于深度神经网络和保守有限差分方案的非线性回归的分裂方案。数值测试在可能的高维框架中显示出非常好的结果。