A numerical algorithm for solving fully nonlinear parabolic equations based on forward-backward stochactic differencial equations and neural networks

The article presents a numerical method for solving the Cauchy problem for a fully nonlinear parabolic equation. Considered equation is reduced to a system of quasilinear parabolic equations. A probabilistic representation of this system solution was constructed based on the solution of the system of forward-backward stochastic differential equations (FBSDE). The FBSDE solution is reduced to solving an optimization problem solved numerically using a neural network. An example of applying this method to an equation describing the price of an optimal portfolio in the Black-Scholes market is considered. The numerical solution was tested while choosing utility functions for which exact solutions exist.