[论文解读] On full history recursive multilevel Picard approximations and numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations

E Weinan, Martin Hutzenthaler+2 more|arXiv (Cornell University)|Jul 12, 2016

Stochastic processes and financial applications被引用 3

一句话总结

该论文提出了一种用于高维非线性抛物型PDE和BSDE的全历史递归多水平Picard（FHRMLP）逼近方法，结合了多水平蒙特卡洛与Picard不动点迭代。证明了计算成本在维度上呈线性增长，在精度倒数上呈四次方增长，实现了对任意 $ \delta > 0 $ 的 $ O(d\epsilon^{-(4+\delta)}) $ 复杂度，且在100维问题中得到了数值验证。

ABSTRACT

Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental tools in the state-of-the-art pricing and hedging of financial derivatives. The PDEs and BSDEs appearing in such applications are often high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs are typically not available, it is a very active topic of research to solve such PDEs and BSDEs approximately. In this paper we introduce a family of new approximation methods for high-dimensional PDEs and BSDEs. A key idea of our methods is to combine multilevel approximations with Picard fixed-point approximations. Thereby we obtain a class of full history recursive multilevel Picard approximations. Our error analysis proves for one of the proposed approximation methods that if $\epsilon\in(0,\infty)$ is the prescribed approximation accuracy and if $d\in\mathbb{N}$ is the dimension of the considered PDE, then for every $\delta \in (0,\infty)$ it holds that the computational effort of the approximation method (number of function evaluations of the coefficient functions of the considered PDE and number of used independent scalar standard normal random variables) is at most $O(d\,\epsilon^{-(4+\delta)})$, that is, the computational effort grows only linearly in the dimension and up to an arbitrarily small order quartically in the reciprocal of the prescribed approximation accuracy. We illustrate the efficiency of one of the proposed approximation methods by means of numerical simulations presenting approximation accuracy against runtime for several nonlinear PDEs from physics (such as the Allen-Cahn equation) and financial engineering (such as derivative pricing incorporating default risks) in the case of $d=100$ space dimensions.

研究动机与目标

• 解决金融工程与物理中出现的高维、非线性抛物型PDE和BSDE的挑战。
• 克服此类方程数值逼近中的维度灾难问题。
• 开发一类新型的多水平Picard逼近方法，递归地利用之前所有迭代的完整历史信息。
• 建立严格的误差界，表明计算成本仅随维度线性增长，随精度倒数多项式增长。
• 通过100维实际方程（如Allen-Cahn方程和违约风险定价）的数值模拟，展示该方法的高效性。

提出的方法

• 为高维PDE和BSDE提出一族全历史递归多水平Picard（FHRMLP）逼近方法。
• 将多水平蒙特卡洛技术与Picard迭代相结合，以降低方差并提升收敛性。
• 通过递归依赖于所有先前迭代结果，提升各层级上的逼近精度。
• 通过路径遍历与期望形式的公式，将该方法应用于确定性PDE和随机BSDE。
• 构建一个逼近层级结构，其中每一层通过独立的随机样本对前一层进行细化。
• 通过统计函数评估次数和标准正态随机变量的使用数量，分析计算复杂度。

实验结果

研究问题

• RQ1 多水平Picard方法能否扩展至全历史递归形式，以在高维PDE中提升收敛性？
• RQ2 全历史递归多水平Picard逼近方法的计算复杂度在维度和精度方面如何？
• RQ3 该方法在来自物理与金融的高维非线性PDE中数值表现如何？
• RQ4 该方法是否实现了计算成本随维度线性增长、随精度倒数多项式增长？
• RQ5 该方法能否有效处理实际方程（如Allen-Cahn方程和违约风险模型）中的非线性和高维性？

主要发现

• 一种FHRMLP方法的计算成本被限制在 $ O(d\epsilon^{-(4+\delta)}) $，对任意 $ \delta > 0 $ 成立，证明了其在维度上的线性增长和在精度倒数上的近似四次方增长。
• 该方法在保持高维问题（包括100维情形）精度的同时实现了该复杂度。
• 数值模拟证实了该方法的高效性，100维非线性PDE中表现出精度与运行时间之间的有利权衡。
• 该方法成功处理了Allen-Cahn方程等非线性PDE以及金融工程中的违约风险BSDE。
• 误差分析证实了其在高维与非线性设定下的鲁棒性，且不依赖于维度的指数增长。
• 通过100维模拟中的运行时间与精度曲线对比，表明该方法在可扩展性方面优于标准方法。