In section 4.6 of his book on probability models, Ross analyzes the gambler ruin problem (a gambler who starts with some units of money, stops when he either reaches N or goes broke). In particular, he provides an expression for the expected number of time units the gambler will spend in each of the possible states. This got me thinking about a rich gambler. So rich, that his fortune is bottomless. This guy can afford to take an infinite amount of debt from his account. So, -1, -2, -3, … are all possibilities for him. He simply keeps tossing his coin till his accumulated fortune reaches N. Now, the important difference is that his state space becomes infinitely large. In Ross's example since the state space was finite, we got:
$$S=(I-P_T)^{-1}$$
(here, $S$ is the matrix of expected time units spent in state $j$ given one starts in state $i$ and $P_T$ is the transition probability matrix).
For the bottomless gambler, this simply becomes an infinite system of equations with no recursively cancelling terms.
$$s_{i,j} = \delta_{i,j} + (1-p)s_{i-1,j}+ps_{i+1,j}$$
The problem obviously has an answer. How do I go about solving it?