Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control

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 type formula applied to $f(t, \X(t))$ where $f:[0,T] \times H \rightarrow \R$ is a $C^{0,1}$ 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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Pré-publication, Document de travail
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Contributeur : Francesco Russo <>
Soumis le : dimanche 12 juin 2016 - 16:08:59
Dernière modification le : vendredi 21 septembre 2018 - 10:54:07
Document(s) archivé(s) le : mardi 13 septembre 2016 - 10:10:40


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  • HAL Id : hal-00720490, version 2
  • ARXIV : 1207.5710



Giorgio Fabbri, Francesco Russo. Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control. 2014. 〈hal-00720490v2〉



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