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We consider multitype Markovian branching processes evolving in a Markovianrandom environment. To determine whether or not the branching process becomesextinct almost surely is akin to computing the maximal Lyapunov exponent of asequence of random matrices, which is a notoriously difficult problem. Wedefine dual processes and we construct bounds for the Lyapunov exponent. Thebounds are obtained by adding or by removing information: to add informationresults in a lower bound, to remove information results in an upper bound andwe show that to add more information gives smaller lower bounds. We give a fewillustrative examples and we observe that the upper bound is generally moreaccurate than the lower bound.

Lyapunov exponents for branching processes in a random environment: The effect of information