  We introduce the class of multistage stochastic optimization problems with a
random number of stages. For such problems, we show how to write dynamic
programming equations and detail the Stochastic Dual Dynamic Programming
algorithm to solve these equations. Finally, we consider a portfolio selection
problem over an optimization period of random duration. For several instances
of this problem, we show the gain obtained using a policy that takes the random
duration of the number of stages into account over a policy built taking a
fixed number of stages (namely the maximal possible number of stages).
