论文标题
深度学习算法用于求解高维非线性后向随机微分方程
Deep learning algorithms for solving high dimensional nonlinear backward stochastic differential equations
论文作者
论文摘要
在这项工作中，我们提出了一种基于深度学习的新方案，用于解决高维非线性后向随机微分方程（BSDES）。这个想法是将问题重新重新制定为包括本地损失功能的全球优化。本质上，我们使用深神网络及其具有自动分化的梯度近似BSDE的未知解。通过在每个时间步骤定义的二次局部损耗函数中最小化近似值来进行近似，该函数始终包括终端条件。通过迭代终端条件的时间积分的Euler离散化来获得这种损失函数。我们的配方可以促使随机梯度下降算法不仅要考虑到每个时间层的准确性，而且还会收敛到良好的局部最小值。为了证明我们的算法的性能，提供了几种高维非线性BSDE，包括金融中的定价问题。
In this work, we propose a new deep learning-based scheme for solving high dimensional nonlinear backward stochastic differential equations (BSDEs). The idea is to reformulate the problem as a global optimization, where the local loss functions are included. Essentially, we approximate the unknown solution of a BSDE using a deep neural network and its gradient with automatic differentiation. The approximations are performed by globally minimizing the quadratic local loss function defined at each time step, which always includes the terminal condition. This kind of loss functions are obtained by iterating the Euler discretization of the time integrals with the terminal condition. Our formulation can prompt the stochastic gradient descent algorithm not only to take the accuracy at each time layer into account, but also converge to a good local minima. In order to demonstrate performances of our algorithm, several high-dimensional nonlinear BSDEs including pricing problems in finance are provided.
