Year: 2012
Type: article
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.
DOI: 10.1214/ecp.v17-1825
Authors: Aleksandar Mijatovic, Mikhail Urusov
Institutions: Imperial College London, University of Duisburg-Essen
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: gold
Estimated APC paid: $675
OpenAlex ID: W2058639296