  We consider the problem of detecting jumps in an otherwise smoothly evolving
trend whilst the covariance and higher-order structures of the system can
experience both smooth and abrupt changes over time. The number of jump points
is allowed to diverge to infinity with the jump sizes possibly shrinking to
zero. The method is based on a multiscale application of an optimal jump-pass
filter to the time series, where the scales are dense between admissible lower
and upper bounds. For a wide class of non-stationary time series models and
trend functions, the proposed method is shown to be able to detect all jump
points within a nearly optimal range with a prescribed probability
asymptotically under mild conditions. For a time series of length $n$, the
computational complexity of the proposed method is $O(n)$ for each scale and
$O(n\log^{1+\epsilon} n)$ overall, where $\epsilon$ is an arbitrarily small
positive constant. Numerical studies show that the proposed jump testing and
estimation method performs robustly and accurately under complex temporal
dynamics.
