[Paper Review] 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 applications3 citations

TL;DR

This paper introduces full history recursive multilevel Picard (FHRMLP) approximations for high-dimensional nonlinear parabolic PDEs and BSDEs, combining multilevel Monte Carlo with Picard fixed-point iterations. It proves that computational cost scales linearly in dimension and quartically in inverse accuracy, achieving $ O(d\epsilon^{-(4+\delta)}) $ complexity for any $ \delta > 0 $, with numerical validation in 100D problems.

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.

Motivation & Objective

• Address the challenge of solving high-dimensional, nonlinear parabolic PDEs and BSDEs arising in financial engineering and physics.
• Overcome the curse of dimensionality in numerical approximation of such equations.
• Develop a new class of multilevel Picard approximations that recursively use full history of previous iterates.
• Establish rigorous error bounds showing computational cost grows only linearly in dimension and polynomially in inverse accuracy.
• Demonstrate the method's efficiency through numerical simulations in 100D settings for real-world equations like Allen-Cahn and default-risk pricing.

Proposed method

• Propose a family of full history recursive multilevel Picard (FHRMLP) approximations for high-dimensional PDEs and BSDEs.
• Integrate multilevel Monte Carlo techniques with Picard iteration to reduce variance and improve convergence.
• Use recursive dependence on all prior iterates to enhance approximation accuracy across levels.
• Apply the method to both deterministic PDEs and stochastic BSDEs via pathwise and expectation-based formulations.
• Construct a hierarchy of approximations where each level refines the previous using independent random samples.
• Analyze computational complexity by counting function evaluations and standard normal random variables used.

Experimental results

Research questions

• RQ1 Can multilevel Picard methods be extended to full history recursion to improve convergence in high-dimensional PDEs?
• RQ2 What is the computational complexity of full history recursive multilevel Picard approximations in terms of dimension and accuracy?
• RQ3 How does the method perform numerically in high-dimensional nonlinear PDEs from physics and finance?
• RQ4 Does the method achieve a computational cost that scales linearly in dimension and polynomially in inverse accuracy?
• RQ5 Can the method handle nonlinearities and high-dimensionalities in real-world equations like Allen-Cahn and default-risk models?

Key findings

• The computational cost of one FHRMLP method is bounded by $ O(d\epsilon^{-(4+\delta)}) $ for any $ \delta > 0 $, proving linear scaling in dimension and near-quartic scaling in inverse accuracy.
• The method achieves this complexity while maintaining accuracy across high-dimensional problems, including 100D cases.
• Numerical simulations confirm the method's efficiency, showing favorable trade-off between accuracy and runtime for 100D nonlinear PDEs.
• The approach successfully handles nonlinear PDEs such as the Allen-Cahn equation and BSDEs with default risk in financial engineering.
• The error analysis confirms robustness under high-dimensional and nonlinear settings, with no exponential dependence on dimension.
• The method outperforms standard approaches in terms of scalability, as demonstrated by runtime vs. accuracy curves in 100D simulations.