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

february, 2014
Publication type:
Lecture note
arXiv:
images/icons/icon_arxiv.png 1207.5710
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.
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 It\^o 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.
BibTeX:
@misc{Fab-Rus-2014,
    title={Infinite dimensional weak Dirichlet processes, stochastic PDEs 
           and optimal control. },
    year={2014 },
    month={2},
    comment={{umatype:'cours'}},
}