A note on a.s. finiteness of perpetual integral functionals of diffusions. (English) Zbl 1111.60061

Summary: We use the boundary classification of diffusions in order to derive a criterion for the convergence of perpetual integral functionals of transient real-valued diffusions. We present a second approach, based on R. Khas’minskij’s lemma [Theor. Probab. Appl. 4, 309–318 (1960); translation from Teor. Veroyatn. Primen. 4, 332–341 (1959; Zbl 0089.34501)], which is applicable also to spectrally negative Lévy processes. In the particular case of transient Bessel processes, our criterion agrees with the one obtained via T. Jeulin’s convergence lemma [in: Séminaire de probabilités XVI. Lect. Notes Math. 920, 248–256 (1982; Zbl 0483.60020)].


60J60 Diffusion processes
60J65 Brownian motion
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