The Inverse Problem for Rough Controlled Differential Equations

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.

Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering

“…We suggest a subsampling-based method, establishing connections to multiscale parameter estimation as investigated in [89]; see Section 2.3. Other approaches towards obtaining ∆Y k have been developed in [9,44,71]. We would like to stress that although estimating ∆Y skew k works reasonably well in our experiments (see Sections 6.1 and 6.2), in many applications it might yield satisfactory results to neglect the skew-symmetric part, that is, to use the approximation ∆Y k = ∆Y sym k , either because the observation path is one-dimensional, or because the Lévy area term is comparatively small (see Section 6.3).…”

“…In other circumstances, setting ∆Y skew k = 0 might provide a reasonable approximation, see Section 6.3. Failing that, our experiments show promising results based on the following procedure (for other approaches towards estimating Lévy areas see [9,44,71]):…”

“…The theory of rough paths provides a principled route towards constructing continuous modifications Φ of Φ (employed, for instance, in [29,37]) by replacing stochastic integrals in terms of rough integrals defined on appropriately lifted paths (Y, Y). Although the difficulty of obtaining or imposing the additional information Y is well known (see, however, [9,44,71] and Section 6 of this article), recent works have demonstrated the potential of including path signatures into data-driven methods to compress information efficiently or unveil multiscale structure [21,76].…”

The Inverse Problem for Controlled Differential Equations

“…Consider the controlled differential equation (CDE) (1) dY t = f (Y t ) • dX t , Y 0 = y 0 , 0 ≤ t ≤ T.…”

“…In other words, how do we solve the inverse problem for CDEs? This has also been studied in [1], but in a different context: the authors reconstruct the truncated signature of the control X on a fixed window from observations of the increments of the solution to the CDE on that window for a number of different initial conditions.…”