2024/10/23 by Dan Crisan, Crisan, Dan, Martin P. Clark +1
Computer Science · #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2410.17737
openalex publication_date 2024/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper studies the identification of an ℝd-valued diffusion X when a running function of it, say h(Xt), is observed. A point-wise observation of the process (in other words, observing h(Xt) in isolation) cannot identify Xt unless the h is injective. However observing h(Xs) on a small interval [t,t+ε] can be enough to determine Xt 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 X0. For real-analytic h the property reduces to simple asymmetry; that is, there is no nontrivial affine isometry κ on ℝd such that h = h ∘ κ. A second result concerns the case where X0 is given and h is merely Borel; then X is adapted to an augmented filtration generated by the observation process (h(Xt))t≥ 0 if h is `locally invertible' on a subset of ℝd dense in the fine topology on ℝd.