This comment is problematic. You're confusing discrete-state-space models with discrete-time models. You can have any combination of continuous/discrete Markov process:

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).