通过签名控制微分方程实现动态通用逼近
Dynamic Universal Approximation via Signature Controlled Differential Equations
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中文总结 AI 辅助
研究签名控制微分方程，发展其存在性等理论并转化为签名泛函条件，证明简单参数化的Sig-CDEs能逼近适定路径依赖CDE的解路径，还通过新空间重铸方程并研究截断方程，建立提升路径依赖动力学的方法。
中文摘要 AI 辅助
我们研究签名控制微分方程（Sig-CDEs），即向量场通过签名映射分解的路径依赖控制微分方程（CDEs）。在停止的赫尔德路径空间上，我们为一般路径依赖CDEs发展了存在性、唯一性和稳定性理论，并将这些路径适定性标准转化为相应签名泛函的条件。然后我们证明了动态通用性：简单参数化的Sig-CDEs能在有界控制集和初始历史上任意好地逼近任何适定路径依赖CDE的解路径，通过加权空间得到全局变体。在这个框架内，类群元素的整个映射提供了一类特定的Sig-CDEs。使用一类新的极限张量空间，我们将Sig-CDEs重铸为无限维经典CDEs，并通过规范型缩放论证证明其适定性，从而建立了提升一般路径依赖动力学的原则方法。最后，我们研究截断的Sig-CDEs作为内在条件下阶梯-N李群上的有限维微分方程，即其适定性根据基础群度量来表述。
英文摘要
We study signature controlled differential equations (Sig-CDEs), that is, path-dependent controlled differential equations (CDEs) whose vector fields factor through the signature map. Working on spaces of stopped Hölder paths, we develop an existence, uniqueness, and stability theory for general path-dependent CDEs, and translate these pathwise well-posedness criteria into conditions on the corresponding signature functionals. We then prove dynamic universality: simply parametrized Sig-CDEs approximate the solution path of any well-posed path-dependent CDE arbitrarily well, uniformly over bounded sets of controls and initial histories, with global variants obtained using weighted spaces. Within this framework, entire maps of group-like elements provide a specific class of Sig-CDEs. Using a new class of limiting tensor spaces, we recast Sig-CDEs as infinite-dimensional classical CDEs and prove their well-posedness via a gauge-type scaling argument, thereby establishing a principled way to lift generic path-dependent dynamics. Lastly, we study truncated Sig-CDEs as finite-dimensional differential equations on step$-N$ Lie groups under intrinsic conditions, that is, with well-posedness formulated in terms of the underlying group metric.
