Estimating models based on Markov jump processes given fragmented observation series

We consider the problem of estimating the rate matrix governing a finitestate Markov jump process given a number of fragmented time series. We propose to concatenate the observed series and to employ the emerging non-Markov process for
estimation.We describe the bias arising if standard methods forMarkov processes are
used for the concatenated process, and provide a post-processing method to correct
for this bias. This method applies to discrete-timeMarkov chains and to more general
models based on Markov jump processes where the underlying state process is not
observed directly. This is demonstrated in detail for a Markov switching model. We
provide applications to simulated time series and to financial market data, where estimators
resulting from maximum likelihood methods and Markov chain Monte Carlo
sampling are improved using the presented correction.