Convergence Analysis of Iterative Deep Learning Algorithms for Fully Nonlinear BSPDEs in Non-Markovian Utility Maximization

Abstract In utility maximization with non-Markovian setting, the value function satisfies a stochastic Hamilton-Jacobi-Bellman (HJB) equation, a fully nonlinear backward stochastic partial differential equation (BSPDE). We propose iterative deep learning algorithms for such BSPDEs and analyze their convergence. We derive the error estimate of the time discretization scheme for BSPDEs, that of the policy iteration scheme for discretized BSPDEs, and that of the iterative deep learning scheme for discretized BSPDEs. We also test the algorithms and show their convergence and accuracy with numerical examples, including Markovian Heston volatility model and non-Markovian rough volatility model.