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A notion of directed information between two continuous-time processes isproposed. A key component in the definition is taking an infimum over allpossible partitions of the time interval, which plays a role no lesssignificant than the supremum over "space" partitions inherent in thedefinition of mutual information. Properties and operational interpretations inestimation and communication are then established for the proposed notion ofdirected information. For the continuous-time additive white Gaussian noisechannel, it is shown that Duncan's classical relationship between causalestimation and information continues to hold in the presence of feedback uponreplacing mutual information by directed information. A parallel result isestablished for the Poisson channel. The utility of this relationship is thendemonstrated in computing the directed information rate between the input andoutput processes of a continuous-time Poisson channel with feedback, where thechannel input process is constrained to be constant between events at thechannel output. Finally, the capacity of a wide class of continuous-timechannels with feedback is established via directed information, characterizingthe fundamental limit on reliable communication.

Directed Information, Causal Estimation, and Communication in Continuous Time