Reproduce “A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations” on OpenResearch

About this paper

Philipp Grohs, Fabian Hornung, Arnulf Jentzen, Philippe von Wurstemberger

This paper presents a rigorous mathematical proof that artificial neural networks overcome the curse of dimensionality in numerically approximating solutions to Black-Scholes partial differential equations. The proof demonstrates that the required number of ANN parameters grows polynomially with both the PDE dimension and the reciprocal of the approximation accuracy, allowing for efficient, high-dimensional function approximation over an entire domain.

Key insights

• Artificial neural networks are theoretically proven to overcome the curse of dimensionality for approximating solutions of Black-Scholes PDEs over an entire domain.
• The required number of ANN parameters scales at most polynomially with respect to both the PDE dimension and the reciprocal of the approximation accuracy.
• The methodology bridges the gap between probabilistic Monte Carlo approximations and deterministic ANN constructions through a careful theoretical framework.

Summary by alphaXiv.