Information processes for semimartingale experiments (Q1872330)

For a homogeneous filtered statistical experiment \((\Omega ,\mathfrak {F},F,\{P_{\theta }\}_{\theta \in \Theta })\) let \(z_{t}(\theta ,Q)\) be the density process with respect to a dominating distribution and \(\alpha \) be a prior on \(\Theta\). As generalization of the Hellinger integral and the Hellinger transform that play a fundamental role in decision theory the authors introduce the geometric mean process \(g_{t}(\alpha ,Q)=\exp \{\int \ln z_{t}(\theta ,Q)\alpha (d\theta)\}\) which is a nonnegative supermartingale of class (D). The Doob-Meyer and the multiplicative decomposition are established. If the observations come from a semimartingale, then these decompositions are given in terms of the triplet of predictable characteristics. This is used to compute the additive and multiplicative decomposition of the information process. Explicit representations for this process and the belonging Kullback-Leibler information are obtained in special cases, such as for point processes, processes with independent increments, time discrete processes, and the fractional Brownian motion.