Deep Learning-Driven Operator Splitting for Free Boundary Problems: Application to American Option Pricing in Incomplete Markets

This study proposes a deep learning-driven operator splitting scheme for high-dimensional free boundary problems, applied to American option pricing in incomplete markets. The pricing problem is a five-dimensional highly nonlinear variational inequality, which the scheme decomposes along time layers into an elliptic continuation equation approximated by a physics-informed neural network and a closed-form projection step enforcing the early exercise constraint. The scheme is validated against finite difference and least-squares Monte Carlo benchmarks, with option prices and learned exercise boundaries closely matching reference solutions.