The Inverse Problem for Single Trajectories of Rough Differential Equations

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 $\delta\to 0$. 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 $\delta$.