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Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control

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Author Info

  • 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.

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File URL: http://sites.uclouvain.be/econ/DP/IRES/2012017.pdf
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Bibliographic Info

Paper provided by Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES) in its series Discussion Papers (IRES - Institut de Recherches Economiques et Sociales) with number 2012017.

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Length: 56
Date of creation: 24 Jul 2012
Date of revision:
Handle: RePEc:ctl:louvir:2012017

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Related research

Keywords: Covariation and Quadratic variation; Calculus via regularization; Infinite dimensional analysis; Tensor analysis; Dirichlet processes; Generalized Fukushima decomposition; Stochastic partial differential equations; Stochastic control theory;

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Cited by:
  1. Cristina Girolami & Giorgio Fabbri & Francesco Russo, 2014. "The covariation for Banach space valued processes and applications," Metrika, Springer, vol. 77(1), pages 51-104, January.

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