Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control

Giorgio Fabbri and Francesco Russo ()
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Francesco Russo: ENSTA ParisTech, Unité de Mathématiques appliquées, Paris (France)

No 2012017, LIDAM Discussion Papers IRES from Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES)

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.

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 (search for similar items in EconPapers)
Pages: 56
Date: 2012-07-24
New Economics Papers: this item is included in nep-ore
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Citations: View citations in EconPapers (3)

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Persistent link: https://EconPapers.repec.org/RePEc:ctl:louvir:2012017

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