Mutual Information for Stochastic Signals and LÉvy Processes

In this paper, some relations between estimation and mutual information are given by expressing two mutual information calculations in terms of two distinct estimation errors. Specifically the mutual information between a stochastic signal and a pure jump Levy process whose rate function depends on the signal is expressed in terms of a filtering error and the rate of change of this mutual information with respect to a parameter multiplying the rate function of the Levy process is expressed in terms of a smoothing error. These results generalize the analogous mutual information results for some Gaussian noise processes with additive stochastic signals.