@article{Dai2024, 
author = {Wanyang Dai},
title = {Simulating a strongly nonlinear backward stochastic partial differential equation via efficient approximation and machine learning},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {7},
pages = {18688-18711},
keywords = {backward stochastic partial differential equation (B-SPDE), Monte Carlo simulation, strongly nonlinear, Cauchy terminal value problem, machine learning (ML), convolutional neural network (CNN)},
url = {https://www.sciopen.com/article/10.3934/math.2024909},
doi = {10.3934/math.2024909},
abstract = {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.}
}