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

This paper proposes a deep learning-driven operator splitting scheme for high-dimensional free-boundary problems, with American option pricing as a representative application. In a market with proportional transaction costs, a stochastic liquidity risk with an empirically validated nonlinear drift based on S&P 500 data, and two-factor stochastic volatility, the pricing problem takes the form of a five-dimensional highly nonlinear variational inequality. The proposed scheme decomposes this variational inequality at each time layer into an elliptic continuation equation approximated by a physics-informed neural network and a closed-form projection step enforcing the early-exercise constraint. A conditional $\ell^\infty$ stability bound is derived under stated resolvent and Lipschitz assumptions. The scheme is validated against finite-difference and least-squares Monte Carlo benchmarks, and both the option prices and the learned exercise boundary show close agreement with the reference solutions. Sensitivity analyses with respect to the key model parameters are also reported.