  Consider the following communication scenario. An encoder observes a
stochastic process and causally decides when and what to transmit about it,
under a constraint on the expected number of bits transmitted per second. A
decoder uses the received codewords to causally estimate the process in real
time. The encoder and the decoder are synchronized in time. For a class of
continuous Markov processes satisfying regularity conditions, we find the
optimal encoding and decoding policies that minimize the end-to-end estimation
mean-square error under the rate constraint. We show that the optimal encoding
policy transmits a $1$-bit codeword once the process innovation passes one of
two thresholds. The optimal decoder noiselessly recovers the last sample from
the 1-bit codewords and codeword-generating time stamps, and uses it to decide
the running estimate of the current process, until the next codeword arrives.
In particular, we show the optimal causal code for the Ornstein-Uhlenbeck
process and calculate its distortion-rate function. Furthermore, we show that
the optimal causal code also minimizes the mean-square cost of a
continuous-time control system driven by a continuous Markov process and
controlled by an additive control signal.
