  We consider multitype Markovian branching processes evolving in a Markovian
random environment. To determine whether or not the branching process becomes
extinct almost surely is akin to computing the maximal Lyapunov exponent of a
sequence of random matrices, which is a notoriously difficult problem. We
define dual processes and we construct bounds for the Lyapunov exponent. The
bounds are obtained by adding or by removing information: to add information
results in a lower bound, to remove information results in an upper bound and
we show that to add more information gives smaller lower bounds. We give a few
illustrative examples and we observe that the upper bound is generally more
accurate than the lower bound.
