Excursions of diffusion processes and continued fractions

Comtet, Alain; Tourigny, Yves

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 3, pp. 850-874.

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.

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.

Classification : 60J60, 30B70
Mots clés : diffusion processes, continued fraction, Riccati equation, excursions, Stieltjes transform

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