Equivalence to uniqueness in the concept of predictability between filtrations

Abstract

In this paper we develop the concept of dependence between filtrations given in [11], named causal predictability, which is based on the Granger?s definition of causality. Then, we provide some new properties of this concept and prove a result that consider equivalence to uniqueness of the given concept. Also, a few examples that illustrate applications of the given concept are given with the main focus on stochastic differential equations (SDE) and financial mathematics. The study of Granger?s causality has been defined in the context of time series. Since continuous time models become more and more frequent in econometric practice, epidemiology, climatology, demographic, etc, we develop a concept connected to the continuous time processes. At the same time, the modern finance theory extensively uses diffusion processes.