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A Stochastic Calculus Approach for the Brownian Snake

Published online by Cambridge University Press:  20 November 2018

Jean-Stéphane Dhersin  and
Laurent Serlet
Jean-Stéphane Dhersin
Affiliation:
UFR de Mathématiques et d’Informatique, Université René Descartes, 45 rue des Saint Pères, 75270 Paris Cedex 06, France, email: dhersin@math-info.univ-paris5.frserlet@math-info.univ-paris5.fr
Laurent Serlet
Affiliation:
UFR de Mathématiques et d’Informatique, Université René Descartes, 45 rue des Saint Pères, 75270 Paris Cedex 06, France, email: dhersin@math-info.univ-paris5.frserlet@math-info.univ-paris5.fr
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Abstract

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We study the “Brownian snake” introduced by Le Gall, and also studied by Dynkin, Kuznetsov, Watanabe. We prove that Itô’s formula holds for a wide class of functionals. As a consequence, we give a new proof of the connections between the Brownian snake and super-Brownian motion. We also give a new definition of the Brownian snake as the solution of a well-posed martingale problem. Finally, we construct a modified Brownian snake whose lifetime is driven by a path-dependent stochastic equation. This process gives a representation of some super-processes.

Keywords

60J2560G4460J8060J60
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 52 , Issue 1 , 01 February 2000 , pp. 92 - 118
Copyright
Copyright © Canadian Mathematical Society 2000

References

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