Filtered statistical models and Hellinger processes

In this paper we extend the notion of Hellinger processes, which was known for pairs of probability measures defined on a filtered space, to general filtered statistical experiments. In a sense, this notion generalizes the Mellin transforms of a (nonfiltered) statistical experiment. Then we characterize several properties of statistical experiments in terms of their Hellinger processes, namely the continuity of the likelihood processes, or the property that the likelihood processes are exponentials of processes with independent increments, or the Gaussian property of likelihoods. We also devote a lot of space to proving that, under mild additional assumptions, a statistical model is "generated" by a process with independent increments if and only if it has deterministic Hellinger processes.