From Irrevocably Modulated Filtrations to Dynamical Equations Over Random Networks

Abstract

We develop a probabilistic information framework via what we call irrevocably modulated filtrations produced by non-invertible matrix-valued jump processes acting on multivariate observation processes carrying noisy signals. Under certain conditions, we provide dynamical representations of conditional expectation martingales in systems where signals from randomly changing information networks may get irreversibly amalgamated or switched-off over random time horizons. We apply the framework to scenarios where the flow of information goes through multiple modulations before reaching observing agents. This leads us to introduce a Lie-type operator as a morphism between spaces of sigma-algebras, which quantifies information discrepancy caused by different modulation sequences. As another example, we show how random graphs can be used to generate irrevocably modulated filtrations that lead to pure noise scenarios. Finally, we construct systems that exhibit gradual decay of additional sources of information through the choice of spectral radii of the modulators.