带受控扩散的抛物型Bellman方程的Howard迭代的全局Sobolev收敛性
Global Sobolev convergence of Howard iteration for parabolic Bellman equations with controlled diffusion
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中文总结 AI 辅助
本文证明带受控扩散的抛物型Bellman方程在一般条件下Howard策略迭代的全局Sobolev收敛，并推广至多维及熵正则化模型，关键是通过残差测度收敛构造唯一强解。
中文摘要 AI 辅助
我们证明了对于具有控制依赖扩散的有限时域Bellman方程，Howard策略迭代具有全局Sobolev收敛性。在一维情形下，该结果适用于在一致椭圆性条件下有界可测系数，无需大折扣、短时域、小扩散扰动或对改进策略的正则性假设。我们还获得了在扩散矩阵结构条件下的多维推广。关键观察是，消失的策略改进迫使Bellman残差在测度意义下收敛；更高的可积性随后产生强收敛并构造出唯一的强解。我们将论证推广到熵正则化模型，包括零温极限，并针对特殊的一维模型建立了二次收敛性。
英文摘要
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.
发表机构
- Nankai University（南开大学）
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