In this paper, we mainly focus on solving high-dimensional stochastic Hamiltonian systems with boundary condition, which is essentially a Forward Backward Stochastic Differential Equation (FBSDE in short), and propose a novel method from the view of the stochastic control. In order to obtain the approximated solution of the Hamiltonian system, we first introduce a corresponding stochastic optimal control problem such that the extended Hamiltonian system of the control problem is exactly what we need to solve, then we develop two different algorithms suitable for different cases of the control problem and approximate the stochastic control via deep neural networks. From the numerical results, comparing with the Deep FBSDE method developed previously from the view of solving FBSDEs, the novel algorithms converge faster, which means that they require fewer training steps, and demonstrate more stable convergences for different Hamiltonian systems.
翻译:在本文中,我们主要侧重于解决具有边界条件的高维随机汉密尔顿系统,这基本上是一个向后向后向式阵列差异(简称FBSDE ), 并从随机控制的角度提出一种新的方法。 为了获得汉密尔顿系统的近似解决方案,我们首先引入了相应的随机最佳控制问题,这样,我们所需要的正是控制问题扩大的汉密尔顿系统,然后我们开发了两种不同的算法,两种不同的算法,分别适合控制问题的不同案例和通过深神经网络接近随机控制的情况。 从数字结果来看,与以前从解决密尔密尔顿系统的角度开发的深FBSDE方法相比较,新算法更快地趋同,这意味着它们需要较少的培训步骤,并显示不同汉密尔顿系统更稳定的趋同。