Optimal Control Under Stochastic Target Constraints
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Thu, 22/01/2009 14:15 |
Bruno Bouchard (Paris, Dauphine) |
Stochastic Analysis Seminar  |
DH 1st floor SR |
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We
study a class of Markovian optimal stochastic control problems in which the
controlled process  is constrained to satisfy an a.s.~constraint
  at some final time  . When the set is of the form  , with  non-decreasing in  , 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
 which characterizes the set ![$ D:=\{(t,Z^\nu(t))\in
[0,T]\x\R^{d+1}~:~Z^\nu(T)\in G\;a.s. $](/files/tex/84beaea64d1ec18b5e14bde09f1aec29b77c05a7.png) for some  . Contrary to standard
state constraint problems, the domain  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 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
 or  . |
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