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Optimal Control under Stochastic Target Constraints

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Date
2010
Dewey
Probabilités et mathématiques appliquées
Sujet
Stochastic Target Problem; discontinuous viscosity solutions; State Constraint Problem; Optimal Control
Journal issue
SIAM Journal on Control and Optimization
Volume
48
Number
5
Publication date
2010
Article pages
3501-3531
DOI
http://dx.doi.org/10.1137/090757629
URI
https://basepub.dauphine.fr/handle/123456789/1961
Collections
  • CEREMADE : Publications
Metadata
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Author
Bouchard, Bruno
Elie, Romuald
Imbert, Cyril
Type
Article accepté pour publication ou publié
Abstract (EN)
We study a class of Markovian optimal stochastic control problems in which the controlled process $Z^\nu$ is constrained to satisfy an a.s.~constraint $Z^\nu(T)\in G\subset \R^{d+1}$ $\Pas$ at some final time $T>0$. When the set is of the form $G:=\{(x,y)\in \R^d\x \R~:~g(x,y)\ge 0\}$, with $g$ non-decreasing in $y$, we provide a Hamilton-Jacobi-Bellman characterization of the associated value function. It gives rise to a state constraint problem where the constraint can be expressed in terms of an auxiliary value function $w$ which characterizes the set $D:=\{(t,Z^\nu(t))\in [0,T]\x\R^{d+1}~:~Z^\nu(T)\in G\;a.s.$ for some $ \nu\}$. Contrary to standard state constraint problems, the domain $D$ is not given a-priori and we do not need to impose conditions on its boundary. It is naturally incorporated in the auxiliary value function $w$ which is itself a viscosity solution of a non-linear parabolic PDE. Applying ideas recently developed in Bouchard, Elie and Touzi (2008), our general result also allows to consider optimal control problems with moment constraints of the form $\Esp{g(Z^\nu(T))}\ge 0$ or $\Pro{g(Z^\nu(T))\ge 0}\ge p$.

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