I am experimenting with Monte Carlo Simulations and I have come up with this interesting problem. Suppose we are generating random values using a Normal distribution with St.Dev = 2 and mean = the last value generated (Markov process), we start at the value 5, but every time we generate a value greater than 9 we start generating random values using a second Normal distribution with St.Dev = 3. If we generate a value greater than 15 or less than 0 we start from 5 again. We want to find the expected value of this random process. Now one way would be to just generate a very large amount of samples, but seeing as this would be impractical if we decide to work with a more complicated process, my question is: What is the smart way to estimate the expected value (also probability distribution and other standard characteristics of this random process).
I have looked into the variations of Monte Carlo like Makrkov Chain Monte Carlo (MCMC). Yet I cannot seem to think of a good approach to solving this problem.
Any advice or sources would be helpful :)
PS I am working in Python, but any reference would be helpful, be it a code implementation in some other language, a theoretical explanation or even just the right term to search for on the Internet.