Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T00:56:36.440Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]@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: [email protected]@math-info.univ-paris5.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1] Dawson, D. A., Measure-valued Markov processes. ´ Ecole d’été de probabilités de Saint-Flour 1991, Lecture Notes in Math. 1541 (1993), 1–260.Google Scholar
[2] Dawson, D. A. and Perkins, E. A., Historical processes. Mem. Amer. Math. Soc. (454) 93 (1991).Google Scholar
[3] Dynkin, E. B., Path processes and historical superprocesses. Probab. Theory Related Fields 90 (1991), 1–36.Google Scholar
[4] Ethier, S. N. and Kurtz, T. G., Markov processes, characterization and convergence. Wiley, New York, 1986.Google Scholar
[5] Feller, W., The parabolic differential equations and the associated semi-groups of transformations. Ann. Math. 55 (1952), 458–519.Google Scholar
[6] Le Gall, J-F., A class of path-valued Markov processes and its applications to superprocesses. Probab. Theory Related Fields 95 (1993), 25–46.Google Scholar
[7] Le Gall, J-F., A path-valued Markov process and its connections with partial differential equations. Proceedings in First European Congress of Mathematics, vol. II, Progr. Math. 120 (1994), 185–212.Google Scholar
[8] Priouret, P., Processus de diffusion et équations différentielles stochastiques. E´ cole d’e´te´ de probabilite´s de Saint-Flour 1973, Lecture Notes in Math. 390 (1974), 37–113.Google Scholar
[9] Revuz, D. and Yor, M., Continuous martingales and Brownian motion. 2nd edn, Springer Verlag, 1995.Google Scholar
[10] Watanabe, S., Branching diffusions (superdiffusions) and random snakes. Trends in Probability and Related Analysis (Taipei, 1996), World Scientific Publishing, 1997, 289–304.Google Scholar
[11] Wentzell, A. D., Infinetesimal characteristics of Markov processes in a function space which describes the past. Theory Probab. Appl. 30 (1985), 625–639.Google Scholar