Mutual information, relative entropy and estimation error in semi-martingale channels

Fundamental relations between information and estimation have been established in the literature for the Gaussian and Poisson channels. In this work, we demonstrate that such relations hold for a much larger family of continuous-time channels. We introduce the family of semi-martingale channels where the channel output is a continuous time semi-martingale, and the channel input modulates the characteristics of the semi-martingale. For these channels, which includes as a special case the Gaussian and Poisson models, we establish new representations relating the mutual information between the channel input and output to an optimal causal filtering loss, thereby unifying and considerably extending results from the Gaussian and Poisson settings. Extensions to the setting of mismatched estimation are also presented where the relative entropy between the laws governing the output of the channel under two different input distributions is equal to the cumulative difference between the estimation loss incurred by using the mismatched and optimal causal filters respectively. The results in this work can be viewed as the continuous-time analogues of recent generalizations for relations between information and estimation for scalar transformations via Lévy channels.