  We consider sequences-indexed by time (discrete stages)-of families of
multistage stochastic optimization problems. At each time, the optimization
problems in a family are parameterized by some quantities (initial states,
constraint levels.. .). In this framework, we introduce an adapted notion of
time consistent optimal solutions, that is, solutions that remain optimal after
truncation of the past and that are optimal for any values of the parameters.
We link this time consistency notion with the concept of state variable in
Markov Decision Processes for a class of multistage stochastic optimization
problems incorporating state constraints at the final time, either formulated
in expectation or in probability. For such problems, when the primitive noise
random process is stagewise independent and takes a finite number of values, we
show that time consistent solutions can be obtained by considering a finite
dimensional state variable. We illustrate our results on a simple dam
management problem.
