New smoothing algorithms based on reversed-time lumped models

Corresponding to a process
x(.)
with a known state model propagating in growing time, we obtain a process
x_{r}(.)
, statistically equivalent to
x(.)
up to second-order properties but with a state model propagating in reversed time. This result is exploited to obtain recursive linear least-squares estimation algorithms that evolve backwards in time. The reversed-time model is shown to be closely related to the system adjoint of the original state model. Some operator-theoretic consequences are also noted.