强可测性与适应逼近及其在非马尔可夫控制中的应用
Strong Measurability and Adapted Approximations with Applications to Non-Markovian Control
- University of Calgary（卡尔加里大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究不可分度量空间映射的强可测性与适应逼近，给出可测性蕴含强可测性的条件，并应用于非马尔可夫随机控制，证明状态和代价的一致收敛及最优值收敛。
AI中文摘要：
我们研究了取值于可能不可分度量空间的映射的强可测性和适应逼近。在回顾度量空间Pettis准则后，我们给出了可测性蕴含强可测性的条件。在连续统假设（CH）下，从可数生成可测空间到度量空间的每个可测映射，对于定义域上的每个测度都是强可测的。对于有限或σ-有限测度，当目标密度严格小于每个实值可测基数时，同样的结论成立。在CH下，这包括密度至多为𝔠的目标；如果不存在实值可测基数，则对所有度量目标成立。对于函数值映射，我们表明逐点可测性和截面的连续性并不必然保证函数空间拓扑中的可测性。我们利用可数个求值和值域的本质上可分性给出了一个验证准则。进一步，我们在适当的滤过假设下获得了渐进或联合可测适应过程的初等L^p逼近，具有路径和Wasserstein变量的Lipschitz泛函，以及基于具有左连续或右连续路径的过程的有限多个观测的光滑逼近。最后，在CH和适当假设下，我们将这些结果应用于可分Hilbert空间中的受控随机微分方程，允许跳跃、随机系数、路径依赖以及状态和控制律依赖；我们建立了在容许控制上状态和代价的一致收敛、最优值的收敛以及近优控制的转移。
英文摘要：
We study strong measurability and adapted approximations for maps taking values in possibly nonseparable metric spaces. After recalling the metric-space Pettis criterion, we give conditions under which measurability implies strong measurability. Under the continuum hypothesis (CH), every measurable map from a countably generated measurable space into a metric space is strongly measurable for every measure on the domain. For finite or $σ$-finite measures, the same conclusion holds when the target density is strictly smaller than every real-valued measurable cardinal. Under CH this includes targets of density at most $\mathfrak c$; if no real-valued measurable cardinal exists, it holds for all metric targets. For function-valued maps, we show that pointwise measurability and continuity of the sections need not ensure measurability in the function-space topology. We provide a verification criterion using countably many evaluations and essential separability of the range. Further, we obtain elementary $L^p$-approximations of progressive or jointly measurable adapted processes under suitable filtration assumptions, Lipschitz functionals with path and Wasserstein variables, and smooth approximations based on finitely many observations of a process with left- or right-continuous paths. Finally, under CH and suitable assumptions, we apply these results to controlled stochastic differential equations in separable Hilbert spaces, allowing for jumps, random coefficients, path dependence, and state- and control-law dependence; we establish convergence of states and costs uniformly over admissible controls, convergence of optimal values, and transfer of near-optimal controls.
