On a gateway between continuous and discrete Bessel and Laguerre processes

Miclo, Laurent; Patie, Pierre

Annales Henri Lebesgue, Volume 2 (2019), pp. 59-98.

By providing instances of approximation of linear diffusions by birth-death processes, Feller [Fel50] has offered an original path from the discrete world to the continuous one. In this paper, by identifying an intertwining relationship between squared Bessel processes and some linear birth-death processes, we show that this connection is in fact more intimate and goes in the two directions. As by-products, we identify some properties enjoyed by the birth-death family that are inherited from squared Bessel processes. For instance, these include a discrete self-similarity property and a discrete analogue of the beta-gamma algebra. We proceed by explaining that the same gateway identity also holds for the corresponding ergodic Laguerre semi-groups. It follows again that the continuous and discrete versions are more closely related than thought before, and this enables to pass information from one semi-group to the other one.

En donnant des exemples d’approximation de diffusions linéaires par des processus de vie et de mort, Feller [Fel50] a mis en évidence un lien entre les mondes discrets et continus. En identifiant une relation d’entrelacement entre les processus de carrés de Bessel et certains processus de vie et de mort linéaires, nous montrons que ces liens sont encore plus profonds et vont également dans les deux sens. Cela nous permet d’exhiber des propriétés originales pour cette famille de processus de vie et de mort directement héritées des processus de carrés de Bessel. En particulier, nous obtenons une propriété d’auto-similarité discrète et un analogue discret à l’algèbre bêta-gamma. Nous expliquons ensuite comment ces propriétés s’étendent aux semi-groupes de Laguerre correspondants. Il apparaît à nouveau que les versions continues et discrètes sont plus proches qu’imaginé précédemment et qu’il est possible de transférer des informations d’un semi-groupe à l’autre.

Classification: 53C28, 53C26, 32Q45
Keywords: Squared Bessel processes, linear birth-death processes, intertwining, continuous and discrete scaling, spectral decomposition.

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