How to write an error model for a first order Markov process

Question

In the context of Kalman filters I’ve come across a text book question which states to write out the error model for a first order Markov process. Whilst I understand that a Markov process is the continuous version of a Markov chain I don’t understand how you can write out an unspecified error model. I can’t find any resources which explain this what a first order Markov process error model is - could someone explain what is meant and/or point me to some resources which explain this?

Answer 1

In this context a "first order Markov process" does not refer to a Markov chain, but rather to a stochastic process sometimes called "coloured noise" which satisfies the time-invariant SDE $x'(t) = \frac1\tau x(t) +q_c(t)$, where $\tau$ is the model time constant and $q_c$ a white noise process. Some relevant results are that the variance of such a process is constant in time, its power spectral density behaves much differently than those of white noise and random walk, and the autocorrelation function decays exponentially with the time shift.

It is indeed not possible to write out an unspecified error model. The error model will, of course, depend on the model in question.