Year: 2025
Type: article
Abstract: This paper studies the identification of an $$\mathbb {R}^d$$ R d -valued diffusion X when a running function of it, say $$h(X_t)$$ h ( X t ) , is observed. A point-wise observation of the process (in other words, observing $$h(X_t)$$ h ( X t ) in isolation) cannot identify $$X_t$$ X t unless the h is injective. However observing $$h(X_s)$$ h ( X s ) on a small interval $$[t,t+\varepsilon ]$$ [ t , t + ε ] can be enough to determine $$X_t$$ 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 $$X_0$$ X 0 . For real-analytic h the property reduces to simple asymmetry; that is, there is no nontrivial affine isometry $$\kappa $$ κ on $$\mathbb {R}^d$$ R d such that $$h = h \circ \kappa $$ h = h ∘ κ . A second result concerns the case where $$X_0$$ X 0 is given and h is merely Borel; then X is adapted to an augmented filtration generated by the observation process $$(h(X_t))_{t\ge 0}$$ ( h ( X t )
Authors: J. M. C. Clark, Dan Crisan
Institution: Imperial College London
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4416715501