英文标题：
《Solving Nonlinear and High-Dimensional Partial Differential Equations
via Deep Learning》
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作者：
Ali Al-Aradi, Adolfo Correia, Danilo Naiff, Gabriel Jardim, Yuri
Saporito
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最新提交年份：
2018
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英文摘要：
In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal execution, mean field games, etc. The main idea behind DGM is to represent the unknown function of interest using a deep neural network. A key feature of this approach is the fact that, unlike other commonly used numerical approaches such as finite difference methods, it is mesh-free. As such, it does not suffer (as much as other numerical methods) from the curse of dimensionality associated with highdimensional PDEs and PDE systems. The main goals of this paper are to elucidate the features, capabilities and limitations of DGM by analyzing aspects of its implementation for a number of different PDEs and PDE systems. Additionally, we present: (1) a brief overview of PDEs in quantitative finance along with numerical methods for solving them; (2) a brief overview of deep learning and, in particular, the notion of neural networks; (3) a discussion of the theoretical foundations of DGM with a focus on the justification of why this method is expected to perform well.
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中文摘要：
在这项工作中，我们应用Sirignano和Spiliopoulos（2018）中描述的深层伽辽金方法（DGM）来解决定量金融应用中出现的许多偏微分方程，包括期权定价、最优执行、平均场博弈等。DGM背后的主要思想是使用深层神经网络来表示未知的兴趣函数。这种方法的一个关键特征是，与其他常用的数值方法（如有限差分法）不同，它是无网格的。因此，它不会（像其他数值方法一样）受到与高维偏微分方程和偏微分方程系统相关的维数灾难的影响。本文的主要目的是通过分析DGM在许多不同PDE和PDE系统中的实现，阐明DGM的特点、功能和局限性。此外，我们还提出：（1）简要概述了定量金融中的偏微分方程及其数值求解方法；（2） 深度学习的简要概述，尤其是神经网络的概念；（3） 讨论了DGM的理论基础，重点讨论了为什么该方法预期表现良好的理由。
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分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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