Approximate Likelihood Construction for Rough Differential Equations

Abstract

In this paper, we propose a new framework for the construction of the likelihood of discretely observed differential equations driven by rough paths. The paper is split in two parts: in the first part, we construct the exact likelihood for a discretely observed rough differential equation, driven by a piecewise linear path. In the second part, we use this likelihood to construct approximate likelihoods for discretely observed differential equations driven by a general class of rough paths. Finally, we study the behaviour of the approximate likelihoods when the sampling frequency tends to infinity.

The Inverse Problem for Controlled Differential Equations

“…This can be of independent interest, allowing, for example, one to make inference about the distribution of the random control X driving the system [9]. It also provides an indirect way of learning the vector field: in the case where the control is a realisation of a random path with known distribution, reconstructing the path X conditioned on the vector field f can lead to the construction of an approximate likelihood [12]. In the context of neural CDEs where the input X is unobserved, there are situations where the same unobserved control X could be applied to a known system, thus making it possible to first infer X and then use existing methods for learning the vector field [8].…”