Particle Methods with Deep Learning for Stochastic Control under Partial Observation

Numerical computation of stochastic control problems under partial observation is challenging because the dynamic programming formulation is naturally posed on the conditional distribution of the hidden state. We propose particle-based methods that replace this infinite-dimensional filtering state by a finite-dimensional weighted particle system, building on recent limit theory for mean-field control with common-noise-adapted controls. We prove, under suitable assumptions, convergence of the fully discretized particle approximation to the original continuous-time partially observed control problem. The particle reformulation is high-dimensional but permutation-invariant, a structure that can be exploited by symmetric neural network architectures. We develop two deep learning algorithms: a direct optimization method for feedback controls and a Deep BSDE method for particle problems admitting a backward stochastic differential equation representation. We also extend the computational framework to partially observed mean-field control problems, which have been studied theoretically but remain less developed numerically. Numerical experiments on a linear--quadratic benchmark, a nonlinear partially observed mean-field control problem, and two financial applications, portfolio liquidation and asset allocation, demonstrate the accuracy and practical utility of the approach.