Differential learning methods for solving fully nonlinear PDEs

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