  We address the output regulation problem for a general class of linear
stochastic systems. Specifically, we formulate and solve the ideal
full-information and output-feedback problems, obtaining perfect, but
non-causal, asymptotic regulation. A characterisation of the problem
solvability is deduced. We point out that the ideal problems cannot be solved
in practice because they unrealistically require that the Brownian motion
affecting the system is available for feedback. Drawing from the ideal
solution, we formulate and solve approximate versions of the full-information
and output-feedback problems, which do not yield perfect asymptotic tracking
but can be solved in a realistic scenario. These solutions rely on two key
ideas: first we introduce a discrete-time a-posteriori estimator of the
variations of the Brownian motion obtained causally by sampling the system
state or output; second we introduce a hybrid state observer and a hybrid
regulator scheme which employ the estimated Brownian variations. The
approximate solution tends to the ideal as the sampling period tends to zero.
The proposed theory is validated by the regulation of a circuit subject to
electromagnetic noise.
