We propose a new multistep deep learning-based algorithm for the resolution of moderate to high dimensional nonlinear backward stochastic differential equations (BSDEs) and their corresponding parabolic partial differential equations (PDE). Our algorithm relies on the iterated time discretisation of the BSDE and approximates its solution and gradient using deep neural networks and automatic differentiation at each time step. The approximations are obtained by sequential minimisation of local quadratic loss functions at each time step through stochastic gradient descent. We provide an analysis of approximation error in the case of a network architecture with weight constraints requiring only low regularity conditions on the generator of the BSDE. The algorithm increases accuracy from its single step parent model and has reduced complexity when compared to similar models in the literature.

翻译：我们提出了一种基于深度学习的新型多步算法，用于求解中等至高维非线性倒向随机微分方程（BSDE）及其对应的抛物型偏微分方程（PDE）。该算法依赖于BSDE的迭代时间离散化，并在每个时间步利用深度神经网络和自动微分逼近其解及梯度。通过在每个时间步借助随机梯度下降法顺序最小化局部二次损失函数，获得近似解。在生成元仅需低正则性条件的权值约束网络架构情形下，我们给出了近似误差分析。该算法较其单步基础模型提升了精度，且与文献中同类模型相比具有更低的复杂度。