  The objective of this paper is to study detectability, observability and
related Lyapunov-type theorems of linear discrete-time time-varying stochastic
systems with multiplicative noise. Some new concepts such as uniform
detectability, ${\cal K}^{\infty}$-exact detectability (resp. ${\cal
K}^{WFT}$-exact detectability, ${\cal K}^{FT}$-exact detectability, ${\cal
K}^{N}$-exact detectability) and ${\cal K}^{\infty}$-exact observability (resp.
${\cal K}^{WFT}$-exact observability, ${\cal K}^{FT}$-exact observability,
${\cal K}^{N}$-exact observability) are introduced, respectively, and nice
properties associated with uniform detectability, exact detectability and exact
observability are also obtained. Moreover, some Lyapunov-type theorems
associated with generalized Lyapunov equations and exponential stability in
mean square sense are presented under uniform detectability, ${\cal
K}^{N}$-exact observability and ${\cal K}^{N}$-exact detectability,
respectively.
