谱Barron空间中带状态依赖扩散的有限时间域Hamilton-Jacobi-Bellman方程
Variable-Coefficient Parabolic Equations and Finite-Horizon Hamilton--Jacobi--Bellman Equations in Augmented Spectral Barron Spaces
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中文总结 AI 辅助
该研究在谱Barron空间中，结合线性理论与半显式梯度迭代推导高维带状态依赖扩散的HJB方程解，建立其与神经网络逼近的关联，为PDE解分析到神经网络复杂度提供直接路径。
中文摘要 AI 辅助
我们研究带一致椭圆型、状态依赖扩散系数的受控扩散过程的高维有限时间域Hamilton-Jacobi-Bellman方程。受需建立严谨分析框架以解释高维随机控制中神经网络逼近的需求驱动，我们在增广谱Barron空间中开展分析。对于变系数线性方程，我们通过在高斯乘子中冻结二阶系数构造参量，得到无需扩散系数空间小变化的精确格林算子与终值传播子。随后，我们将该线性理论与非线性HJB方程的半显式梯度迭代结合，证明短时间域收敛性，其极限为有界经典解，且通过伊藤验证论证将其与随机控制价值函数对应。最后，我们推导时空联合浅余弦网络逼近。综上，我们的分析将高维随机控制、变系数抛物型正则性、非线性HJB理论与定量神经网络逼近相连接，从而提供了从PDE解分析到神经网络复杂度的直接路径。
英文摘要
We establish a whole-space solution framework in augmented spectral Barron spaces for uniformly elliptic parabolic equations with state-dependent principal coefficients. A frozen-symbol parametrix yields bounded Green and terminal operators and a one-derivative smoothing estimate on arbitrary finite horizons, without requiring the spatial variation of the principal coefficient to be perturbatively small. We then apply this linear framework to a finite-horizon Hamilton--Jacobi--Bellman equation. A semi-explicit gradient iteration converges on sufficiently short horizons at a Gamma-factorial rate in a whole-space augmented Barron norm, and its limit is identified with the stochastic-control value function. Finally, temporal Jackson approximation and spatial spectral Barron approximation give joint shallow cosine-network approximations in space and time for the value function and the optimal feedback, with quantitative error and neuron-count bounds. Taken together, these results connect variable-coefficient parabolic well-posedness and smoothing in augmented spectral Barron spaces with nonlinear HJB solvability and quantitative neural-network approximation.
发表机构
- Zhongtai Securities Institute for Financial Studies, Shandong University（山东大学中泰证券金融研究院）
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