I am currently working on linear birth death processes with transition probabilities g(n,n+1) = nb and g(n,n-1) = nd.
Lets consider the probabilities of the stopping times $$P(t_N < t_0| X_0 = n) =: p_N(n)$$ and from that we can get the boundary value problem
$$nb p_N(n+1) + nd p_N(n-1) = n(d+b)p_N(n).$$
with p_N(N) = 1 and p_N(0) = 0.
I have a problem to see how we get this equation. I thought of using the total probability and since the linear birth death process is only jumping to state n+1 or n-1 we get $$ p_N(n) = d/(d+b) p_N(n-1) + b/(d+b)p_N(n+1).$$
Is this correct? I saw this approach somewhere else but I am not really sure because for me the definition of the law of total probability is $P(A) = P(A |B)P(B) + P(A|B^*)P(B^*).$
I would appreciate any hint or help! Thank you very much!