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# Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control

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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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Cited literature [65 references]

https://hal.inria.fr/hal-00720490
Contributor : Francesco Russo Connect in order to contact the contributor
Submitted on : Sunday, June 12, 2016 - 4:08:59 PM
Last modification on : Wednesday, May 11, 2022 - 12:06:06 PM
Long-term archiving on: : Tuesday, September 13, 2016 - 10:10:40 AM

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Stoch-Verif27fev2014FR.pdf
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### Identifiers

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

### Citation

Giorgio Fabbri, Francesco Russo. Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control. {date}. ⟨hal-00720490v2⟩

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