  Optimal zero-delay coding (quantization) of a vector-valued Markov source
driven by a noise process is considered. Using a stochastic control problem
formulation, the existence and structure of optimal quantization policies are
studied. For a finite-horizon problem with bounded per-stage distortion
measure, the existence of an optimal zero-delay quantization policy is shown
provided that the quantizers allowed are ones with convex codecells. The
bounded distortion assumption is relaxed to cover cases that include the linear
quadratic Gaussian problem. For the infinite horizon problem and a stationary
Markov source the optimality of deterministic Markov coding policies is shown.
The existence of optimal stationary Markov quantization policies is also shown
provided randomization that is shared by the encoder and the decoder is
allowed.
