Concepts of Statistical Causality and Strong and Weak Properties of Predictable Representation

The paper considers the statistical concept of causality in continuous time, which is based on Granger’s definition of causality. We give necessary and sufficient conditions, in terms of statistical causality, for the preservation of the strong property of predictable representation for stopped martingales when filtration is decreased. This concept of causality is also connected to the preservation of the strong property of predictable representation under a change in measure. In addition, we give conditions, in terms of statistical causality, for martingales to have strong and weak properties of predictable representation. The results are applied to the problem of pricing claims in incomplete financial markets.