Finite-Horizon Hamilton--Jacobi--Bellman Equations with State-Dependent Diffusion in Spectral Barron Spaces
We study high-dimensional finite-horizon Hamilton--Jacobi--Bellman equations for controlled diffusions with uniformly elliptic, state-dependent diffusion coefficients. Motivated by the need for a rigorous analytical framework that explains neural-network approximation in high-dimensional stochastic control, we formulate the analysis in the augmented spectral Barron space. For the variable-coefficient linear equation, we construct a parametrix by freezing the second-order coefficient in the Gaussian multiplier, leading to an exact Green operator and terminal propagator without requiring small spatial variation of the diffusion coefficient. We then combine this linear theory with a semi-explicit gradient iteration for the nonlinear HJB equation and prove short-horizon convergence. The limit is a bounded classical solution and is identified with the stochastic-control value function by an It\^o verification argument. Finally, we derive a joint shallow cosine-network approximation in space and time. Taken together, our analysis connects high-dimensional stochastic control, variable-coefficient parabolic regularity, and nonlinear HJB theory with quantitative neural-network approximation, thereby providing a direct route from PDE solution analysis to neural-network complexity.