I need a stochastic process book which works on semi markov processes in details. Do you have any suggestions?Thanks

I am doing a PhD on regime-jumping diffusion processes, where the intra-jump times are of any distribution, and the regime-to-regime jumps (i to j) are Markov, hence the process is semi Markov.Well, there is very little; I've had to develop a lot of the material myself. Ross's book Stochastic Processes has some good material, see esp. Ch. 8Then a book (out of print) by Cinlar published in 1975. I did not gain much from this book.There is a book specializing on reliability and semi-Markov processes (Semi-Markov Processes and Reliability by Gheorghe Oprisan); this book was not that useful.What are using the semi-Markov model for?

HandlerDo you know where I can find the reference for Cox process? How is Ross' book?

What Ross does is develop differential Kolmogorov equations for semi-Markov processes in an accessible way. That's it really.The emphasis is on a differential equation model: instead of N Markov states, we have, under Ross's model, an infinite number of states where each new state is a pair, viz. the true embedded-Markov-chain state i, and the sojourn time of the process in i.Hope this helps. I am not famliar with Cox processes but if I remember correctly, there was no citation to them as such.Just recalled a martingale description of semi-Markov processes in IEEE Transactions on Automatic Contol; can look it up if there's interest.

Semi-Markov processes are widely used for queuing theory. In this application, the inter-arrivals are not necessarily exponential(and therefore not Markovian), but you can construct a sequence of stopping times so that the process sampled on these optionaltime becomes Markov. Any stochastic process books on queuing theory will contain some chapters on this. Cinlar is one of'em.There are graduate level text books on this area too.Cox process comes from the Cox regression model on mortality. Books on survival analysis or counting processes are what youare looking for. One of best (theoretical) references is "Statistical models based on counting processes", by Anderson, Borgan, Gill, Keiding.Working on credit model?