On the Relation of One-Dimensional Diffusions on Natural Scale and Their Speed Measures

David Criens ()
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David Criens: Albert-Ludwigs University of Freiburg

Journal of Theoretical Probability, 2023, vol. 36, issue 4, 2339-2358

Abstract: Abstract It is well known that the law of a one-dimensional diffusion on natural scale is fully characterized by its speed measure. Stone proved a continuous dependence of such diffusions on their speed measures. In this paper we establish the converse direction, i.e., we prove a continuous dependence of the speed measures on their diffusions. Furthermore, we take a topological point of view on the relation. More precisely, for suitable topologies, we establish a homeomorphic relation between the set of regular diffusions on natural scale without absorbing boundaries and the set of locally finite speed measures.

Keywords: Diffusion; Speed measure; Homeomorphism; Convergence of diffusions; Sufficient and necessary conditions; Limit theorem; Vague convergence; Weak convergence; Feller–Dynkin property; Itô diffusion; 60J60; 60G07; 60F17 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-023-01247-8

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