Finite time stochastic stability and the analysis of tracking systems

A (Liapunov-like) method is presented for obtaining upper bounds of the probability
P_{x}{\sup_{T\geq t\geq 0} V(X_{t})\geq\lambda}
, where
x_{0} = x
and x
t
is a Markov process with either discrete or continuous parameter, and
V(\cdot)
is some function. Such estimates are the quantity of greatest interest in numerous tracking, control, and reliability studies. The method involves finding suitable (stochastic) Liapunov functions. The results are also results in (what may be termed) finite-time stochastic stability. The theorems are based on some theorems of Dynkin [1]. Several illustrative examples are given.