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2012 Convergence of integral functionals of one-dimensional diffusions
Aleksandar Mijatovic, Mikhail Urusov
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Electron. Commun. Probab. 17: 1-13 (2012). DOI: 10.1214/ECP.v17-1825


In this paper we describe the pathwise behaviour of the integral functional $\int_0^t f(Y_u)\,du$ for any $t\in[0,\zeta]$, where $\zeta$ is (a possibly infinite) exit time of a one-dimensional diffusion process $Y$ from its state space, $f$ is a nonnegative Borel measurable function and the coefficients of the SDE solved by $Y$ are only required to satisfy weak local integrability conditions. Two proofs of the deterministic characterisation of the convergence of such functionals are given: the problem is reduced in two different ways to certain path properties of Brownian motion where either the Williams theorem and the theory of Bessel processes or the first Ray-Knight theorem can be applied to prove the characterisation. As a simple application of the main results we give a short proof of Feller's test for explosion.<br />


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Aleksandar Mijatovic. Mikhail Urusov. "Convergence of integral functionals of one-dimensional diffusions." Electron. Commun. Probab. 17 1 - 13, 2012.


Accepted: 16 December 2012; Published: 2012
First available in Project Euclid: 7 June 2016

zbMATH: 1372.60044
MathSciNet: MR3005734
Digital Object Identifier: 10.1214/ECP.v17-1825

Primary: 60H10
Secondary: 60J60

Keywords: Bessel process , Integral functional , Local time , One-dimensional diffusion , Ray-Knight theorem , Williams theorem


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