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Looking around I have found lots of material on continuous time Markov processes on finite or countable state spaces, say $\{0,1,\ldots,J\}$ for some $J\in\mathbb{N}$ or just $\mathbb{N}$. Similarly I have earlier worked with (discrete time) Markov chains on general state spaces, following the modern classic by Meyn & Tweedie.

My question concerns monographs on continuous time Markov processes on general state spaces, say some subset of $\mathbb{R}^k$, $k\in\mathbb{N}$. Are there any good references - preferably but not necessarily suited for an ambitious master student - on this topic?

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  • $\begingroup$ Then, it sounds like you need to write this monograph. $\endgroup$ – Michael Oct 8 '16 at 2:24
  • $\begingroup$ As a first cut, if you assume the transition rates $\nu(x)$ out of each state $x$ are bounded, so that there are values $\nu_{min}, \nu_{max}$ such that $0 < \nu_{min} \leq \nu(x) \leq \nu_{max} < \infty$ for all $x \in \mathcal{S}$, then the embedded chain is a discrete time chain (on a general state space $\mathcal{S}$) and it seems that recurrence properties of the embedded discrete time chain would carry over to the continuous time version. The avg number of jumps to a set of states, as measured in the embedded chain, cannot differ too much from the fraction of time being in those states. $\endgroup$ – Michael Oct 8 '16 at 3:18
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Another classic in probability theory is William Feller's An Introduction to Probability Theory and its Applications, volume I and II, which has a decent coverage of the topic, perhaps it is worth looking up (more specifically chapter 10, vol. II), even though it is a bit old.

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I just finished a course on stochastic calculus, so I still lack experience on continuous time markov processes, but I think I can give you some references I have looked at:

-Chapter 10 of Introduction to Stochastic Integration by Hui-Hsiung Kuo explains the Markov property (and gives examples in terms of the Brownian motion), and diffusion processes. If you have a background in stochastic calculus (this book or Oksendal's are good introductory books) I would start here.

-A more detailed exposition (but too advanced for me at the moment of writing) is the two volume book of Rogers and Williams, Diffussions, Markov Processes and Martingales.

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  • $\begingroup$ I know stochastic calculus. I was looking for expositions which did not limit themselves to diffusion processes. $\endgroup$ – Furrer Oct 7 '16 at 9:29

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