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A functional Itō-formula for Dawson–Watanabe superprocesses

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  • Mandler, Christian
  • Overbeck, Ludger

Abstract

We derive an Itō-formula for the Dawson–Watanabe superprocess, a well-known class of measure-valued processes, extending the classical Itō-formula with respect to two aspects. Firstly, we extend the state–space of the underlying process (X(t))t∈[0,T] to an infinite-dimensional one — the space of finite measure. Secondly, we extend the formula to functions F(t,Xt) depending on the entire paths Xt=(X(s∧t))s∈[0,T] up to times t. This later extension is usually called functional Itō-formula. Finally we remark on the application to predictable representation for martingales associated with superprocesses.

Suggested Citation

  • Mandler, Christian & Overbeck, Ludger, 2022. "A functional Itō-formula for Dawson–Watanabe superprocesses," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 202-228.
    • Handle: RePEc:eee:spapps:v:144:y:2022:i:c:p:202-228
    DOI: 10.1016/j.spa.2021.11.003
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    References listed on IDEAS

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    1. Levental, Shlomo & Schroder, Mark & Sinha, Sumit, 2013. "A simple proof of functional Itô’s lemma for semimartingales with an application," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 2019-2026.
    2. Dawson, D. A., 1975. "Stochastic evolution equations and related measure processes," Journal of Multivariate Analysis, Elsevier, vol. 5(1), pages 1-52, March.
    3. El Karoui, Nicole & Roelly, Sylvie, 1991. "Propriétés de martingales, explosion et représentation de Lévy--Khintchine d'une classe de processus de branchement à valeurs mesures," Stochastic Processes and their Applications, Elsevier, vol. 38(2), pages 239-266, August.
    4. Alexander Schied, 2012. "A control problem with fuel constraint and Dawson-Watanabe superprocesses," Papers 1207.5809, arXiv.org, revised Dec 2013.
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    Cited by:

    1. Gess, Benjamin & Wu, Zhengyan & Zhang, Rangrang, 2026. "Higher order fluctuation expansions for nonlinear stochastic heat equations in singular limits," Stochastic Processes and their Applications, Elsevier, vol. 193(C).

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