A Pontryagin-Guided Neural Policy Optimization Framework for Merton's Portfolio Problem

We present a neural policy optimization framework for Merton's portfolio optimization problem that is rigorously aligned with Pontryagin's Maximum Principle (PMP). Our approach employs a discrete-time, backpropagation-through-time (BPTT)-based gradient method, but unlike conventional data-driven methods, we establish a direct connection to the underlying continuous-time optimality conditions. By approximating adjoint variables from a policy-fixed backward stochastic differential equation (BSDE), we obtain parameter gradients consistent with the PMP framework, all without explicitly solving the PMP-derived BSDE. As the policy parameters are iteratively updated, both the suboptimal adjoint variables and the neural network policies converge almost surely to their PMP-optimal counterparts. This ensures that the final learned policy is not only numerically robust but also provably optimal in the continuous-time sense. Hence, our method provides a theoretically grounded and practically implementable solution that bridges modern deep learning techniques and classical optimal control theory in stochastic settings.