Lyapunov exponents for branching processes in a random environment: The   effect of information
  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. 
Sophie Hautphenne, Guy Latouche
math.PR