Differential learning methods for solving fully nonlinear PDEs

TL;DR

This paper introduces deep learning algorithms that solve fully nonlinear PDEs by combining dual stochastic control representations, neural network-based feedback control estimation, and novel loss functions to improve derivative approximations, especially second derivatives.

Contribution

It presents new deep learning methods that incorporate differential loss functions and Malliavin derivatives to enhance the accuracy of PDE solution derivatives, including second derivatives.

Findings

Accurate solution of nonlinear PDEs in finance and portfolio optimization

Enhanced derivative estimation, especially second derivatives

Effective approximation of families of PDEs with varying terminal conditions

Abstract

We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control representation form, and the corresponding optimal feedback control is estimated using a neural network. Next, three different methods are presented to approximate the associated value function, i.e., the solution of the initial PDE, on the entire space-time domain of interest. The proposed deep learning algorithms rely on various loss functions obtained either from regression or pathwise versions of the martingale representation and its differential relation, and compute simultaneously the solution and its derivatives. Compared to existing methods, the addition of a differential loss function associated to the gradient, and augmented training sets with Malliavin derivatives of the forward process, yields a better estimation of the PDE's solution derivatives, in particular of the second derivative, which is usually difficult to approximate. Furthermore, we leverage our methods to design algorithms for solving families of PDEs when varying terminal condition (e.g. option payoff in the context of mathematical finance) by means of the class of DeepOnet neural networks aiming to approximate functional operators. Numerical tests illustrate the accuracy of our methods on the resolution of a fully nonlinear PDE associated to the pricing of options with linear market impact, and on the Merton portfolio selection problem.