I was wondering if anyone has some well organized material that goes through the relationships among the most important stochastic processes like Markov, Ito, Levy, Martingales etc.
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These online lecture notes and two textbooks listed in this stats.exchange discussion, one of which is free online, are a good introduction to stochastic processes. They also include information on Markov processes, and the lecture notes briefly mentions the Levy property. A shorter introduction to the processes is here.
The above do not detail advanced processes in particular Ito and Martingales. MIT has a great collection of notes on advanced stochastic processes, including Ito and Martingales.