[Paper Review] Multi-level Picard approximations of high-dimensional semilinear parabolic differential equations with gradient-dependent nonlinearities

Martin Hutzenthaler, Thomas Kruse|arXiv (Cornell University)|Nov 3, 2017

Stochastic processes and financial applicationsEconomics, Econometrics and Finance10 references41 citations

TL;DR

This paper introduces a multi-level Picard iteration method combined with Gauss-Legendre quadrature to approximate high-dimensional semilinear parabolic PDEs with gradient-dependent nonlinearities. Under suitable regularity conditions, it proves that the computational complexity grows polynomially in both dimension d and inverse accuracy ε, specifically bounded by O(dε^{-(4+δ)}) for any δ > 0, solving a long-standing open problem in high-dimensional PDE approximation.

ABSTRACT

Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) have a wide range of applications. In particular, high-dimensional PDEs with gradient-dependent nonlinearities appear often in the state-of-the-art pricing and hedging of financial derivatives. In this article we prove that semilinear heat equations with gradient-dependent nonlinearities can be approximated under suitable assumptions with computational complexity that grows polynomially both in the dimension and the reciprocal of the accuracy.

Motivation & Objective

• To address the computational challenge of approximating high-dimensional semilinear parabolic PDEs with gradient-dependent nonlinearities.
• To resolve the open problem of whether such PDEs can be approximated with computational complexity growing polynomially in both dimension d and inverse accuracy ε.
• To extend existing Picard-based methods to handle gradient-dependent nonlinearities, which previous methods could not treat under general terminal conditions.
• To establish a rigorous complexity bound for the proposed method, ensuring scalability in high dimensions.

Proposed method

• Employs multi-level Picard iterations to iteratively approximate the solution of the semilinear PDE.
• Integrates Gauss-Legendre quadrature rules to efficiently approximate time integrals in the Picard iteration, leveraging their spectral convergence for smooth integrands.
• Uses a recursive error bound (inequality 54) to control the global error across levels of the multi-level scheme.
• Applies discrete Gronwall-type inequalities and seminorm estimates to manage error propagation in the presence of gradient-dependent nonlinearities.
• Derives a non-recursive global error bound (inequality 65) by iterating the recursive error estimate.
• Employs stochastic representation of the PDE solution via backward stochastic differential equations (BSDEs) and Monte Carlo sampling for numerical evaluation.

Experimental results

Research questions

• RQ1 Can high-dimensional semilinear parabolic PDEs with gradient-dependent nonlinearities be approximated with computational complexity that grows polynomially in both dimension d and inverse accuracy ε?
• RQ2 Does the multi-level Picard method with Gauss-Legendre quadrature maintain polynomial complexity when the nonlinearity depends on the solution’s gradient?
• RQ3 Is the computational cost of the proposed method bounded by O(dε^{-(4+δ)}) for any δ > 0 under standard regularity assumptions on the solution?
• RQ4 Can the method handle general terminal conditions without requiring them to be small, unlike the branching diffusion method?
• RQ5 What is the precise dependence of the computational complexity on the dimension d and the accuracy ε in the gradient-dependent case?

Key findings

• The computational complexity of the multi-level Picard approximation is bounded by O(dε^{-(4+δ)}) for any δ > 0, where d is the dimension and ε is the prescribed accuracy.
• The method achieves this complexity bound under suitable regularity assumptions on the exact solution, including C∞ smoothness of the solution u∞.
• The use of Gauss-Legendre quadrature ensures fast convergence of the time integrals, which is critical for maintaining low complexity.
• The error analysis establishes a non-recursive global error bound (inequality 65) that enables the complexity estimate.
• The method is applicable to general terminal conditions and does not require the nonlinearity to be small, unlike the branching diffusion method.
• The result confirms that the curse of dimensionality is overcome for this class of PDEs using the proposed multi-level Picard approach.