Solving the Kolmogorov PDE by means of deep learning

TL;DR

This paper introduces a deep learning-based numerical method to solve high-dimensional Kolmogorov PDEs across entire regions, overcoming the curse of dimensionality and improving accuracy and speed in complex models.

Contribution

The paper presents a novel deep learning approach that efficiently approximates solutions to high-dimensional Kolmogorov PDEs over regions, not just at single points, addressing key limitations of existing methods.

Findings

Effective in high dimensions for various models

Overcomes curse of dimensionality in PDE solving

Demonstrates high accuracy and computational speed

Abstract

Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs, respectively, can typically not be solved explicitly and it has been and still is an active topic of research to design and analyze numerical methods which are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nearly all approximation methods for Kolmogorov PDEs in the literature suffer under the curse of dimensionality or only provide approximations of the solution of the PDE at a single fixed space-time point. In this paper we derive and propose a numerical approximation method which aims to overcome both of the above mentioned drawbacks and intends to deliver a numerical approximation of the Kolmogorov PDE on an entire region $[a,b]^d$ without suffering from the curse of dimensionality. Numerical results on examples including the heat equation, the Black-Scholes model, the stochastic Lorenz equation, and the Heston model suggest that the proposed approximation algorithm is quite effective in high dimensions in terms of both accuracy and speed.