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Penalization of Galton–Watson processes

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  • Abraham, Romain
  • Debs, Pierre

Abstract

We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton–Watson processes with a penalizing function of the form P(x)sx where P is a polynomial of degree p and s∈[0,1]. We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s=1 (or s→1) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton–Watson tree with p distinguished infinite spines.

Suggested Citation

  • Abraham, Romain & Debs, Pierre, 2020. "Penalization of Galton–Watson processes," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3095-3119.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:3095-3119
    DOI: 10.1016/j.spa.2019.09.005
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    Cited by:

    1. Abraham Romain & Boulal Sonia & Debs Pierre, 2024. "Penalization of Galton–Watson Trees with Marked Vertices," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3688-3724, November.

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