I am interested in some general theorems related to lower bounds on discrete time finite Markov chains hitting probabilities (preferably ergodic chains , but not necessarily ), with references . Similar results related to continuous time Markov chains, that allow discretization, are welcome. Thank you.
Chapter 10 in the book "Markov chains and mixing times (see http://www.ams.org/publications/authors/books/postpub/mbk-107 and https://pages.uoregon.edu/dlevin/MARKOV/mcmt2e.pdf ) is all about bounding hitting times for discrete chains.
Wouldn't gambler's ruin work? It has a lower bound (being bankrupt), and it is discrete finite MC. You can calculate the probability of being bankrupt too.