单轨迹的粗糙微分方程逆问题
The Inverse Problem for Single Trajectories of Rough Differential Equations
AI总结：
本文研究了单轨迹粗糙微分方程的逆问题，提出通过构造几何p-粗糙路径X，使其响应Y驱动方程匹配观测轨迹y，并发展了离散逆问题的求解框架，证明了在p-变分下的收敛性。
摘要：
受开发一般框架进行离散观测随机粗糙微分方程统计推断的需要，我们的目标是构造一个几何p-粗糙路径X，其响应Y驱动粗糙微分方程时能匹配观测轨迹y。我们称其为连续逆问题，并首先严格定义其解。然后发展了一个框架，使解可以作为适当设计的离散逆问题解的极限，从而在p-变分下收敛。我们的方法基于校准那些定义粗糙路径X到观测轨迹y的有界变分路径。此外，我们开发了一个通用的数值算法来构造离散逆问题的解。核心思想是利用路径的签名表示，迭代响应和控制之间，每次根据所需性质进行修正。我们应用该框架到几何p-粗糙路径X作为p-变分拓扑下分段线性路径极限的情况。我们表达固定观测率下的离散逆问题为由分段线性路径驱动的方程组解，并证明当观测时间δ→0时收敛到连续逆问题的解。最后，我们证明在该背景下，求解离散逆问题的数值算法简化为局部梯度的迭代同时更新，并证明其在δ下关于p-变分的一致收敛性。
英文摘要：
Motivated by the need to develop a general framework for performing statistical inference for discretely observed random rough differential equations, our aim is to construct a geometric $p$-rough path ${\bf X}$ whose response $Y$, when driving a rough differential equation, matches the observed trajectory $y$. We call this the \textit{continuous inverse problem} and start by rigorously defining its solution. We then develop a framework where the solution can be constructed as a limit of solutions to appropriately designed \textit{discrete inverse problems}, so that convergence holds in $p$-variation. Our approach is based on calibrating the bounded variation paths whose limit defines the rough path `lift' of path $X$ to rough path ${\bf X}$ to the observed trajectory $y$. Moreover, we develop a general numerical algorithm for constructing the solution to the discrete inverse problem. The core idea of the algorithm is to use the signature representation of the path, iterating between the response and the control, each time correcting according to the required properties. We apply our framework to the case where the geometric $p$-rough path ${\bf X}$ is defined as the limit of piecewise linear paths in the $p$-variation topology. We express the discrete inverse problem for a fixed observation rate as a solution to a system of equations driven by piecewise linear paths and prove convergence to the solution of the continuous inverse problem for observation time $δ\to 0$. Finally, we show that, in this context, the numerical algorithm for solving the discrete inverse problem simplifies to an iterative simultaneous update of the local gradients and we prove that it converges in $p$-variation uniformly with respect to $δ$.
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