随机神经网络用于大致波动
Antoine Jacquier1,2, Žan Žurič1
1Department of Mathematics, Imperial College London, London, UK.
概括
我们开发了一个深度学习算法来解决复杂的金融数学问题. 这种新的储存神经网络方法为粗略的波动性建模提供了强大而理论上健全的方法.
科学领域:
- 量化金融 量化金融
- 计算数学 计算数学 计算数学
- 机器学习 机器学习
背景情况:
- 路径依赖的部分微分方程 (PDEs) 在金融建模中至关重要,特别是对于粗略波动.
- 通过分析来解决这些复杂的方程往往是难以解决的.
- 现有的数值方法可能会面临高维度和粗略波动动态的挑战.
研究的目的:
- 开发一种基于深度学习的新型数值算法,用于在粗略的波动性建模中解决路径依赖的PDEs.
- 利用最近在随机微分方程和神经网络架构方面的进展.
- 提供理论上有基础的,计算上高效的解决方案.
主要方法:
- 解释部分微分方程 (PDE) 作为一个逆向随机微分方程 (BSDE) 的解.
- 使用一种水库类型的神经网络架构,灵感来自Gonon,Grigoryeva和Ortega.
- 将优化问题用简单的最小平方回归来表达.
主要成果:
- 提出的深度学习算法有效地解决了与粗波动相关的路径依赖的PDEs.
- 储库神经网络方法简化了对最小平方回归问题的优化.
- 对于这个问题,我们建立了储库方法的理论收性质.
结论:
- 深度学习,特别是储备神经网络,为解决复杂的金融PDE提供了强大的工具.
- 与水库网络相结合的BSDE解释提供了一种融合且高效的数值方法.
- 这种方法推进了用于粗略波动性建模的可用的计算技术.
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