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

TL;DR

This paper investigates the extinction probabilities of multitype Markovian branching processes in random environments by analyzing Lyapunov exponents, introducing bounds based on information levels, and demonstrating their effectiveness through examples.

Contribution

It introduces a novel method of bounding Lyapunov exponents using information manipulation, providing practical tools for analyzing complex branching processes.

Findings

Adding information yields tighter lower bounds.

Removing information produces upper bounds that are generally less accurate.

Examples illustrate the bounds' effectiveness in different scenarios.

Abstract

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.