Excursions of diffusion processes and continued fractions
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 3, p. 850-874

It is well-known that the excursions of a one-dimensional diffusion process can be studied by considering a certain Riccati equation associated with the process. We show that, in many cases of interest, the Riccati equation can be solved in terms of an infinite continued fraction. We examine the probabilistic significance of the expansion. To illustrate our results, we discuss some examples of diffusions in deterministic and in random environments.

Il est bien connu que les excursions d'un processus de diffusion peuvent être étudiées en considérant une certaine équation de Riccati associée au processus. On montre que, dans beaucoup de cas intéressants, certaines solutions de cette équation de Riccati peuvent être développées en fraction continue. On examine le contenu probabiliste de ce développement. Ces résultats sont illustrés par quelques exemples de diffusions en milieux aléatoires et déterministes.

DOI : https://doi.org/10.1214/10-AIHP390
Classification:  60J60,  30B70
Keywords: diffusion processes, continued fraction, Riccati equation, excursions, Stieltjes transform
@article{AIHPB_2011__47_3_850_0,
     author = {Comtet, Alain and Tourigny, Yves},
     title = {Excursions of diffusion processes and continued fractions},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {3},
     year = {2011},
     pages = {850-874},
     doi = {10.1214/10-AIHP390},
     zbl = {1266.60138},
     mrnumber = {2841077},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_3_850_0}
}
Comtet, Alain; Tourigny, Yves. Excursions of diffusion processes and continued fractions. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 3, pp. 850-874. doi : 10.1214/10-AIHP390. http://www.numdam.org/item/AIHPB_2011__47_3_850_0/

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