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Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control

Author

Listed:
  • Giorgio FABBRI

    (Centre d’Etudes des Politiques Economiques de l’Université d’Evry, Evry (France) and Dipartimento di Studi Economici S. Vinci, University of Naples Parthenope, Naples (Italy) and IRES, Université catholique de Louvain, Louvain-La-Neuve,)

  • Francesco RUSSO

    (ENSTA ParisTech, Unité de Mathématiques appliquées, Paris (France))

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 Hilbert space H, is the sum of a local martingale and a suitable orthogonal process. The new concept is shown to be useful in several contexts and directions. On one side, the mentioned decomposition appears to be a substitute of an Itô’s type formula applied to f(t;X(t)) where f : [0;T] x H → R is a C0;1 function and, on the other side, the idea of weak Dirichlet process fits the widely used notion of mild solution for stochastic PDE. As a specific application, we provide a verification theorem for stochastic optimal control problems whose state equation is an infinite dimensional stochastic evolution equation.

Suggested Citation

  • 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).
    • Handle: RePEc:ctl:louvir:2012017
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    Cited by:

    1. 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.
    2. 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.

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