Solving the Kolmogorov PDE by means of deep learning

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.

33 pages, 1 figure Accepted for publication in the Journal of Scientific Computing

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Cited by in corpus (11)

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• Deep splitting method for parabolic PDEs
• The Modern Mathematics of Deep Learning
• An overview on deep learning-based approximation methods for partial differential equations
• Convergence analysis for gradient flows in the training of artificial neural networks with ReLU activation
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• An energy-based deep splitting method for the nonlinear filtering problem
• Numerical computation of probabilities for nonlinear SDEs in high dimension using Kolmogorov equation