Before revision: Asset prices contain information about the probability distribution of future states and the stochastic discounting of these states . Without additional assumptions, probabilities and stochastic discounting cannot be separately identified. To understand this identification challenge , we extract a positive martingale component from the stochastic discount factor process using Perron-Frobenius Theory.
After revision: Asset prices contain information about the probability distribution of future states and the stochastic discounting of those states as used by investors. To better understand the challenge in distinguishing investors' beliefs from risk-adjusted discounting, we use Perron-Frobenius Theory to isolate a positive martingale component from the stochastic discount factor process using Perron-Frobenius Theory.