AI 中文总结
该研究针对受控非线性滤波问题，通过集中式N粒子控制问题近似，在特定条件下证明了不同维度下的值函数收敛速率，为相关滤波问题的粒子近似提供了定量误差保证。
AI 中文摘要
我们估计受控非线性滤波问题的粒子近似的值函数收敛速率。状态是平坦环面上的McKean-Vlasov扩散过程，由隐藏的特质噪声和可观测的公共噪声驱动。滤波器（给定观测值的状态条件律）作为控制问题的状态变量，其关联的值函数在Wasserstein空间上求解二阶Hamilton-Jacobi-Bellman方程。我们通过具有独立特质噪声和公共观测噪声的集中式N粒子控制问题来近似该问题，该框架可容纳不可分离的奖励和受控漂移。由于单个控制应用于整个群体，哈密顿量通过群体积分后的优化定义。在数据光滑性、一致椭圆性及该哈密顿量正则性的条件下，我们建立了一致的值函数误差界：d=1时为N^{-1/6}，d=2时为N^{-1/6}(log N)^{1/3}，d>2时为N^{-1/(3d)}。证明结合了公共噪声方向的平移提升、Fourier-Wasserstein下确界与上确界卷积、粘性比较及与N无关的粒子导数估计。
英文摘要
We estimate convergence rates of value functions for particle approximations of a controlled nonlinear filtering problem. The state is a McKean--Vlasov diffusion on the flat torus, driven by hidden idiosyncratic noise and observed common noise. The filter---the conditional law of the state given the observations---serves as the state variable of the control problem, and the associated value function solves a second-order Hamilton--Jacobi--Bellman equation on the Wasserstein space. We approximate this problem by a centralized \(N\)-particle control problem with independent idiosyncratic noises and a common observation noise. The framework accommodates nonseparable rewards and controlled drifts. Since a single control is applied to the entire population, the Hamiltonian is defined by an optimization performed after integration over the population. Under smoothness of the data, uniform ellipticity, and regularity of this Hamiltonian, we establish uniform value-function error bounds of order \(N^{-1/6}\) for \(d=1\), \(N^{-1/6}(\log N)^{1/3}\) for \(d=2\), and \(N^{-1/(3d)}\) for \(d>2\). The proof combines a translation lift in the common-noise direction, Fourier--Wasserstein inf- and sup-convolutions, viscosity comparison, and particle derivative estimates uniform in \(N\).
