A Numerical Algorithm for Solving the Cauchy Problem for a Two-Dimensional Fully Nonlinear Parabolic Equation

The article continues the research presented in􀁢[1]. We solve the Cauchy problem for a two-dimensional fully nonlinear equation arising in the optimization of portfolio investments in the Heston market. First, the fully nonlinear equation is included in a system of quasilinear parabolic equations. The solution of the Cauchy problem for this system is reduced to a stochastic problem in terms of forward-backward stochastic differential equations (FBSDE). Next, the FBSDE is transformed into a stochastic optimal control problem. Finally, this problem is solved using a neural network approach. The proposed numerical algorithm was validated on an example for which it is possible to obtain a quasi-explicit solution.