深度BSVIEs参数化和基于学习的应用程序
1Department of Mathematics, KTH Royal Institute of Technology, Stockholm, 100 44, Sweden.
概括
本研究介绍了一种新的数字方法,用于逆位随机伏尔特拉积分方程 (BSVIEs),这对于记忆金融建模至关重要. 深度学习方法准确地近似这些复杂方程及其反映变量的解决方案.
科学领域:
- 数字分析 数字分析
- 随机过程是指随机的过程.
- 在金融领域的机器学习.
背景情况:
- 倒向随机伏尔特拉积分方程 (BSVIEs) 对于模拟复杂的金融场景,如时间不一致和路径依赖偏好至关重要.
- 现有的方法与BSVIEs固有的二维时间结构和复杂的依赖关系作斗争.
研究的目的:
- 为BSVIEs及其反映的扩展开发一个强大的数值近似框架.
- 在产品概率空间中为BSVIEs建立一个有利位置和可测量的基础.
- 将基于深度学习的倒向随机微分方程 (BSDE) 解决方案扩展到 BSVIE 设置.
主要方法:
- 开发了BSVIE定位性和可测量的框架,使用一个参数化的逆向随机方程家族.
- 引入了一种离散时间学习方案,将哈马古奇-塔古奇离散化与深度神经网络相结合.
- 一般化深度BSDE解决器技术,以解决BSVIEs的二维时间结构.
主要成果:
- 建立了一个严格的对拟议的深度学习方案的融合分析,应用于BSVIEs.
- 成功扩展了数值解析器来处理反射BSVIEs,使得在延迟递归公用事业等领域的应用成为可能.
- 证明了该方法在接近复杂金融模型的解决方案中的有效性.
结论:
- 拟议的深度学习方法为解决BSVIEs及其反映变体提供了一种有效和准确的数值方法.
- 这项工作弥合了理论BSVIE框架和实际计算解决方案之间的差距.
- 这些发现对定量金融有重大影响,特别是在模拟递归公用事业和具有记忆效应的金融衍生工具方面.
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