Electronic Communications in Probability

Convergence of integral functionals of one-dimensional diffusions

Aleksandar Mijatovic and Mikhail Urusov

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Abstract

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 />

Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 61, 13 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465263194

Digital Object Identifier
doi:10.1214/ECP.v17-1825

Mathematical Reviews number (MathSciNet)
MR3005734

Zentralblatt MATH identifier
1372.60044

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60J60: Diffusion processes [See also 58J65]

Keywords
Integral functional one-dimensional diffusion local time Bessel process Ray-Knight theorem Williams theorem

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Mijatovic, Aleksandar; Urusov, Mikhail. Convergence of integral functionals of one-dimensional diffusions. Electron. Commun. Probab. 17 (2012), paper no. 61, 13 pp. doi:10.1214/ECP.v17-1825. https://projecteuclid.org/euclid.ecp/1465263194


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