论文标题
向后深的BSDE方法和非线性问题的应用
Backward Deep BSDE Methods and Applications to Nonlinear Problems
论文作者
论文摘要
在本文中，我们提出了一种用于向后的随机微分方程（FBSDE）的向后深度BSDE方法，其成熟度具有给定的终端条件，该方程在成熟时，将BSDE向后置。我们将此方法应用于非线性定价问题 - 差异率问题。为了使BSDE向后计时，需要解决非线性问题。对于差异率问题，我们得出了这个时步问题和基于泰勒的近似的精确解决方案。以前向后深的BSDE方法仅处理零或线性发生器。虽然先前提到过针对非线性发电机的泰勒方法，但尚未实施或应用，而我们将方法应用于非线性生成器并得出细节并提供结果。同样，以前向后的深度BSDE方法仅针对固定的初始风险因子值$ x_0 $介绍，而我们展示了一个随机$ x_0 $的版本，以及一个在中间时间学习投资组合值的版本。该方法能够在高维度中解决非线性FBSDE问题。
In this paper, we present a backward deep BSDE method applied to Forward Backward Stochastic Differential Equations (FBSDE) with given terminal condition at maturity that time-steps the BSDE backwards. We present an application of this method to a nonlinear pricing problem - the differential rates problem. To time-step the BSDE backward, one needs to solve a nonlinear problem. For the differential rates problem, we derive an exact solution of this time-step problem and a Taylor-based approximation. Previously backward deep BSDE methods only treated zero or linear generators. While a Taylor approach for nonlinear generators was previously mentioned, it had not been implemented or applied, while we apply our method to nonlinear generators and derive details and present results. Likewise, previously backward deep BSDE methods were presented for fixed initial risk factor values $X_0$ only, while we present a version with random $X_0$ and a version that learns portfolio values at intermediate times as well. The method is able to solve nonlinear FBSDE problems in high dimensions.
