Discrete states, discrete time: text, etc.
Discrete states, continuous time: Poisson counting process
Continuous states, discrete time: gambling earnings
Continuous states, continuous time: Brownian motion
Of course, the term "Markov chain" is restricted to discrete time. But you have said "Markov model", which includes all processes with the Markov property (i.e., they forget old state). The model in the OP was not originally a chain, because the radioactive decay process is continuous-time, but he made it into one.
When you say that Markov chains are discretized differential equations, it tempts the reader to think "discretized in time" -- but that's called a difference equation. I think you mean it's discretized in state. I.e., there is a flow of mass from one state to another that is smooth in a diff. eq., and stochastic in the Markov chain. But that's just guesswork.
Also, I would not say a martingale is a discrete random walk. Your statement here introduces the same confusion. Like Markov processes, martingales can be continuous or discrete in both state-space and time (e.g., Brownian motion is a martingale).
I think of a martingale as a mathematical model for a fair game (expected future earnings is zero given available information).
After some further reading , I also agree re: martingales. To be honest I have only ever encountered them in the common discreet-time situation.
http://en.wikipedia.org/wiki/Markov_chain http://en.wikipedia.org/wiki/Continuous-time_Markov_chain
You're completely ignoring immensely important topics like MCMC which (can) operate on uncountable spaces. The vast, vast majority of MCMC research and usage is with continuous random variables.
I'd like to make a point to help the author out, since this post is in the spirit of simple explanations: a good way of thinking of Markov models is that they are fundamentally a discreet form of differential equations, and are useful when we have finitely few states that are generally:
a) easily enumerable b) countably small enough that it doesn't make sense to just replace it with a continuous model
This is similar to the way in which martingales are discreet representations of random walks. Sadly, our education system does a bad job of discreet math, and its typically left to stats courses -- so everything has a different name and few people draw the connections, IMO!
Cheers