  We consider a general stochastic input-output dynamical system with output
evolving in time as the solution to a functional coefficients, It\^{o}'s
stochastic differential equation, excited by an input process. This general
class of stochastic systems encompasses not only the classical communication
channel models, but also a wide variety of engineering systems appearing
through a whole range of applications. For this general setting we find
analogous of known relationships linking input-output mutual information and
minimum mean causal and non-causal square errors, previously established in the
context of additive Gaussian noise communication channels. Relationships are
not only established in terms of time-averaged quantities, but also their
time-instantaneous, dynamical counterparts are presented. The problem of
appropriately introducing in this general framework a signal-to-noise ratio
notion expressed through a signal-to-noise ratio parameter is also taken into
account, identifying conditions for a proper and meaningful interpretation.
