The inverse problem for rough controlled differential equations

TL;DR

This paper establishes a precise condition under which a rough control can be reconstructed from the observed evolution of a controlled differential equation, with applications in stochastic filtering and statistics.

Contribution

It introduces a necessary and sufficient condition for reconstructing rough controls from the controlled evolution, advancing understanding in rough differential equations.

Findings

Reconstruction condition for rough controls established

Applications demonstrated in stochastic filtering and statistics

Practical relevance confirmed through examples

Abstract

We provide a necessary and sufficient condition for a rough control driving a differential equation to be reconstructable, to some order, from observing the resulting controlled evolution. Physical examples and applications in stochastic filtering and statistics demonstrate the practical relevance of our result.