The identification of diffusions from imperfect observations

TL;DR

This paper explores how to identify an $ ext{R}^d$-valued diffusion process from imperfect observations of a function of it, establishing conditions under which the process can be reconstructed from small-interval observations.

Contribution

It introduces the concept of 'fine total asymmetry' for functions $h$, providing necessary and sufficient conditions for diffusion identification from partial observations.

Findings

Observation on small intervals can determine the process at a point.

Fine total asymmetry of $h$ is key for process identification.

For real-analytic $h$, asymmetry ensures identifiability.

Abstract

This paper studies the identification of an $\mathbb{R}^d$-valued diffusion $X$ when a running function of it, say $h(X_t)$, is observed. A point-wise observation of the process (in other words, observing $h(X_t)$ in isolation) cannot identify $X_t$ unless the $h$ is injective. However observing $h(X_s)$ on a small interval $[t,t+\varepsilon]$ can be enough to determine $X_t$ exactly. The paper contain results that expand on this idea; in particular, a property of `fine total asymmetry' of twice continuously differentiable $h$ is introduced that depends on the fine topology of potential theory and that is both necessary and sufficient for $X$ to be adapted to a natural right-continuous filtration generated by the observations. This particular filtration, though augmented with null sets, does not depend on the distribution of $X_0$. For real-analytic $h$ the property reduces to simple asymmetry; that is, there is no nontrivial affine isometry $\kappa$ on $\mathbb{R}^d$ such that $h = h \circ \kappa$. A second result concerns the case where $X_0$ is given and $h$ is merely Borel; then $X$ is adapted to an augmented filtration generated by the observation process $(h(X_t))_{t\geq 0}$ if $h$ is `locally invertible' on a subset of $\mathbb{R}^d$ dense in the fine topology on $\mathbb{R}^d$.