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Penalization for birth and death processes. (English) Zbl 1149.60056
Summary: 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 a 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.
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J25 Continuous-time Markov processes on general state spaces
60J55 Local time and additive functionals
60G44 Martingales with continuous parameter
60G46 Martingales and classical analysis
60J60 Diffusion processes
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[1] Debs, P.: Pénalisation de la marche aléatoire standard par une fonction du maximum unilatère, du temps local en zéro et de la longueur des excursions. Preprint IECN (2007)
[2] Feller, W.: An Introduction to Probability Theory and its Applications, vol. 1, 3rd edn. Wiley, New York (1970) · Zbl 0077.12201
[3] Gradstein, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. Academic, New York (1980)
[4] Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York (1991) · Zbl 0734.60060
[5] Klebaner, F.C.: Introduction to Stochastic Calculus with Applications, 2nd edn. Imperial College Press, London (2005) · Zbl 1077.60001
[6] Lawler, G.: Introduction to Stochastic Processes, 2nd edn. Chapman & Hall/CRC, Boca Raton (2006) · Zbl 1105.60003
[7] Le Gall, J.-F.: Une approche élémentaire des théorémes de décomposition de Williams. In: Séminaire de Probabilités, XX, 1984/85. Lecture Notes in Math., vol. 1204, pp. 447–464. Springer, Berlin (1986)
[8] Mishchenko, A.S.: A discrete Bessel process and its properties. Theory Probab. Appl. 50, 700–709 (2006) · Zbl 1116.60045
[9] Norris, J.: Markov Chains, 2nd printing. Cambridge University Press, Cambridge (1998) · Zbl 0938.60058
[10] Resnick, S.I.: Adventures in Stochastic Processes, 4th printing. Birkhäuser, Boston (2005)
[11] Roynette, B., Yor, M.: Penalising Brownian Paths: Rigorous Results and Meta-Theorems. Astérisque (2007, to appear)
[12] Roynette, B., Vallois, P., Yor, M.: Pénalisations et quelques extensions du théorème de Pitman, relatives au mouvement brownien et à son maximum unilatère. In: Séminaire de Probabilités XXXVIII: In memoriam Paul-André Meyer. Lecture Notes in Math., vol. 1874, pp. 305–336. Springer, Berlin (2006) · Zbl 1124.60034
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