A Policy Iteration Scheme for Semilinear Stochastic Hamilton–Jacobi–Bellman Equations with Exponential Convergence

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The paper addresses non-Markovian stochastic optimal control, which has theoretical relevance to portfolio optimization with random coefficients and incomplete information. However, it is purely theoretical with no numerical experiments, no specific financial applications demonstrated, and no computational framework provided. The SHJB equations arise in stochastic control problems relevant to finance (e.g., utility maximization with random market parameters), but the paper does not directly address trading strategies, risk management, or market microstructure. The methodology could potentially be applied to continuous-time portfolio optimization under regime-switching or random volatility models, but this connection is not explored.

Extremely high complexity. Requires solving a sequence of linear backward stochastic partial differential equations (BSPDEs) in Sobolev spaces. Each iteration involves: (1) computing spatial gradients of the value function, (2) solving a Hamiltonian maximization problem, (3) solving a linear BSPDE with random coefficients. The non-Markovian setting means the value function is a random field, not a deterministic function, requiring sophisticated stochastic analysis tools. No reference implementation or numerical scheme is provided. Practical implementation would require combining the policy iteration framework with existing BSPDE solvers (Monte Carlo, finite element, or deep learning methods).

This is a purely theoretical mathematics paper with no numerical experiments, code, or datasets. All results are analytical proofs of convergence. Reproducibility would require independently verifying the mathematical proofs. No computational implementation is provided.