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, 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.

Based on joint work with W. Lefebvre (LPSM) and G. Loeper (BNP-PAR).

Bio: Huyên Pham is a professor at the University of Paris (Paris Diderot), an adjunct professor at ENSAE, and the chair of applied mathematics at JVN HCM in Vietnam.

Professor Pham’s research interests include stochastic analysis, stochastic control, mean-field theory, large deviations, and machine learning with applications in finance and energy markets. Professor Pham has published more than 200 papers and he is the author of the book ``Continuous-time Stochastic Control and Optimization with Financial Applications.'' Professor Pham is a co-Editor in Chief of Applied Mathematics and Optimization and he is currently serving on the editorial board of various journals including SIAM Journal on Control and Optimization, Mathematical Finance, Risks, Probability, Uncertainty and Quantitative Risk, Acta Mathematica Vietnamica, and Finance and Stochastics.

Professor Pham was awarded the Louis Bachelier Prize by the French Academy of Sciences in 2007. He was elected as a fellow at the University Institute of France in 2006 and a recipient of the Europlace Institute of Finance Prize in 2004.

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