Simulating a strongly nonlinear backward stochastic partial differential equation via efficient approximation and machine learning

We have studied a strongly nonlinear backward stochastic partial differential equation (B-SPDE) through an approximation method and with machine learning (ML)-based Monte Carlo simulation. This equation is well-known and was previously derived from studies in finance. However, how to analyze and solve this equation has remained a problem for quite a long time. The main difficulty is due to the singularity of the B-SPDE since it is a strongly nonlinear one. Therefore, by introducing new truncation operators and integrating the machine learning technique into the platform of a convolutional neural network (CNN), we have developed an effective approximation method with a Monte Carlo simulation algorithm to tackle the well-known open problem. In doing so, the existence and uniqueness of a 2-tuple adapted strong solution to an approximation B-SPDE were proved. Meanwhile, the convergence of a newly designed simulation algorithm was established. Simulation examples and an application in finance were also provided.