Penalization for birth and death processes

Pierre Debs (1, 2) , Mihai Gradinaru (1, 3)

1 IECN - Institut Élie Cartan de Nancy
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
3 IRMAR - Institut de Recherche Mathématique de Rennes

In this paper we study a transient birth and death Markov process penalized by its sojourn time in 0. Under the new probability measure the original process behaves as a recurrent birth and death Markov process. We also show, in a particular case, that an initially recurrent birth and death process, behaves as an transient birth and death process after penalization with the event that it can reach zero in infinite time. We illustrate some of our results with the Bessel random walk example.

Mots clés: Birth and death Markov processes; penalization; sojourn time; Dynkin's formula; random walk; Brownian motion with drift; Bessel chain and process; change of probability

Domaines: Probabilités [math.PR]

Pierre Debs, Mihai Gradinaru. Penalization for birth and death processes. Journal of Theoretical Probability, 2008, 21 (3), pp.745-771. ⟨10.1007/s10959-007-0123-9⟩. ⟨hal-00142589⟩