Global Sobolev convergence of Howard iteration for parabolic Bellman equations with controlled diffusion

RESUMO

We prove global Sobolev convergence of Howard policy iteration for finite-horizon Bellman equations with control-dependent diffusion. In one dimension, the result holds for bounded measurable coefficients under uniform ellipticity, without a large discount, a short horizon, a small diffusion perturbation, or regularity assumptions on the improving policies. We also obtain multidimensional extensions under structural conditions on the diffusion matrix. The key observation is that vanishing policy improvements force the Bellman residual to converge in measure; higher integrability then yields strong convergence and constructs the unique strong solution. We extend the argument to entropy-regularized models, including zero-temperature limits, and establish quadratic convergence for special one-dimensional models.