In my measure-theoretic probability course we covered what the professor called "discrete-time stochastic calculus". Essentially, it was a three part method for computing certain quantities such as expectations of hitting times, optimal stopping times, and certain related probabilities. The process was:
- Find a (sub/super)martingale that yields the appropriate value when stopped.
- Find a Markov process that will give us a difference equation with boundary values/initial values.
- Solve the difference equation to ultimately get the quantity of interest.
I'm looking for a good book that covers this method or methods like this. What area of stochastic analysis does this fall under? Where can I get a resource with good practice problems?