论文标题
七联盟方案：大时的深度学习步骤蒙特卡洛模拟随机微分方程
The Seven-League Scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations
论文作者
论文摘要
我们提出了一个准确的数据驱动的数值方案，以通过大量时间步骤来求解随机微分方程（SDE）。 SDE离散化是通过准确确定的随机搭配（SC）点通过多项式混乱扩展方法来构建的。通过使用人工神经网络来学习这些SC点，我们可以执行大量时间步骤的蒙特卡洛模拟。错误分析证实，只要学习方法稳健且准确，这种数据驱动的方案就可以从强大的融合意义上产生准确的SDE解决方案。使用一种称为压缩压缩搭配和插值技术的方法变体，我们可以大大减少必须学习的神经网络函数的数量，从而提高了计算速度。数值实验在使用大时段时证实了高质量的强收敛误差，而新颖方案的表现优于某些经典的数值SDE离散化。还提出了一些在财务期权估值中的应用程序。
We propose an accurate data-driven numerical scheme to solve Stochastic Differential Equations (SDEs), by taking large time steps. The SDE discretization is built up by means of a polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Error analysis confirms that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a method variant called the compression-decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. Numerical experiments confirm a high-quality strong convergence error when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented.
