  We introduce generalizations of the COGARCH model of Kl\"uppelberg et al.
from 2004 and the volatility and price model of Barndorff-Nielsen and Shephard
from 2001 to a Markov-switching environment. These generalizations allow for
exogeneous jumps of the volatility at times of a regime switch. Both models are
studied within the framework of Markov-modulated generalized Ornstein-Uhlenbeck
processes which allows to derive conditions for stationarity, formulas for
moments, as well as the autocovariance structure of volatility and price
process. It turns out that both models inherit various properties of the
original models and therefore are able to capture basic stylized facts of
financial time-series such as uncorrelated log-returns, correlated squared
log-returns and non-existence of higher moments in the COGARCH case.
