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Infinite Dimensional Weak Dirichlet Processes and Convolution Type Processes

Author

Listed:
  • Giorgio Fabbri

    (GREQAM - Groupement de Recherche en Économie Quantitative d'Aix-Marseille - EHESS - École des hautes études en sciences sociales - AMU - Aix Marseille Université - ECM - École Centrale de Marseille - CNRS - Centre National de la Recherche Scientifique)

  • Francesco Russo

    (OC - Optimisation et commande - UMA - Unité de Mathématiques Appliquées - ENSTA Paris - École Nationale Supérieure de Techniques Avancées)

Abstract

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Banach space H, is the sum of a local martingale and a suitable orthogonal process. The concept of weak Dirichlet process fits the notion of convolution type processes, a class including mild solutions for stochastic evolution equations on infinite dimensional Hilbert spaces and in particular of several classes of stochastic partial differential equations (SPDEs). In particular the mentioned decomposition appears to be a substitute of an Itô's type formula applied to to f(t, X(t)) where f : [0, T ] × H → R is a C0,1 function and X a convolution type processes.

Suggested Citation

  • Giorgio Fabbri & Francesco Russo, 2017. "Infinite Dimensional Weak Dirichlet Processes and Convolution Type Processes," Post-Print halshs-01309384, HAL.
  • Handle: RePEc:hal:journl:halshs-01309384
    DOI: 10.1016/j.spa.2016.06.010
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-01309384
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    References listed on IDEAS

    as
    1. Gozzi, Fausto & Russo, Francesco, 2006. "Weak Dirichlet processes with a stochastic control perspective," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1563-1583, November.
    2. Gozzi, Fausto & Russo, Francesco, 2006. "Verification theorems for stochastic optimal control problems via a time dependent Fukushima-Dirichlet decomposition," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1530-1562, November.
    3. Cristina Girolami & Giorgio Fabbri & Francesco Russo, 2014. "The covariation for Banach space valued processes and applications," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 77(1), pages 51-104, January.
    4. Fabbri, Giorgio & Russo, Francesco, 2017. "Infinite dimensional weak Dirichlet processes and convolution type processes," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 325-357.
    5. Errami, Mohammed & Russo, Francesco, 2003. "n-covariation, generalized Dirichlet processes and calculus with respect to finite cubic variation processes," Stochastic Processes and their Applications, Elsevier, vol. 104(2), pages 259-299, April.
    6. Giorgio FABBRI & Francesco RUSSO, 2012. "Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control," LIDAM Discussion Papers IRES 2012017, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    7. Editors The, 2007. "From the Editors," Basic Income Studies, De Gruyter, vol. 2(1), pages 1-5, June.
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    Cited by:

    1. Giorgio Fabbri & Francesco Russo, 2017. "HJB Equations in Infinite Dimension and Optimal Control of Stochastic Evolution Equations via Generalized Fukushima Decomposition," AMSE Working Papers 1704, Aix-Marseille School of Economics, France.
    2. Giorgio Fabbri & Francesco Russo, 2016. "Infinite Dimensional Weak Dirichlet Processes and Convolution Type Processes," Working Papers halshs-01309384, HAL.
    3. Fabbri, Giorgio & Russo, Francesco, 2017. "Infinite dimensional weak Dirichlet processes and convolution type processes," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 325-357.
    4. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.

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