The Inverse Problem for Single Trajectories of Rough Differential Equations

Abstract. 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 [Formula: see text]-rough path [Formula: see text] whose response [Formula: see text], when driving a rough differential equation, matches the observed trajectory [Formula: see text]. We call this the 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 discrete inverse problems, so that convergence holds in [Formula: see text]-variation. Our approach is based on calibrating the bounded variation paths whose limit defines the rough path ‘lift’ of path [Formula: see text] to rough path [Formula: see text] to the observed trajectory [Formula: see text]. 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 [Formula: see text]-rough path [Formula: see text] is defined as the limit of piecewise linear paths in the [Formula: see text]-variation topology. This is a model assumption that includes some of the most common random rough path families, such as fractional Brownian motion with Hurst parameter [Formula: see text], as the sequence of its nested dyadic piecewise linear interpolations converge almost surely in [Formula: see text]-variation, for [Formula: see text]. 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 [Formula: see text]. 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 [Formula: see text]-variation uniformly with respect to [Formula: see text]. We discuss how we can improve the efficiency of the algorithm by considering splitting methods and reducibility conditions, and we demonstrate the results through a series of numerical examples that highlight the advantages of our algorithm in terms of robustness and efficiency.