  Regime-switching models, in particular Hidden Markov Models (HMMs) where the
switching is driven by an unobservable Markov chain, are widely-used in
financial applications, due to their tractability and good econometric
properties. In this work we consider HMMs in continuous time with both constant
and switching volatility. In the continuous-time model with switching
volatility the underlying Markov chain could be observed due to this stochastic
volatility, and no estimation (filtering) of it is needed (in theory), while in
the discretized model or the model with constant volatility one has to filter
for the underlying Markov chain. The motivations for continuous-time models are
explicit computations in finance. To have a realistic model with unobservable
Markov chain in continuous time and good econometric properties we introduce a
regime-switching model where the volatility depends on the filter for the
underlying chain and state the filtering equations. We prove an approximation
result for a fixed information filtration and further motivate the model by
considering social learning arguments. We analyze its relation to the switching
volatility model and present a convergence result for the discretized model. We
then illustrate its econometric properties by considering numerical
simulations.
