Scale function of a birth-death chain
= Scale function of a birth-death chain
A strictly increasing sequence $s_i$ for which $s(X_t)$ is a <martingale> along the <birth-death chain> before hitting its boundary. With upward and downward probabilities $p_i,q_i$, harmonicity means $p_i(s_{i+1}-s_i)=q_i(s_i-s_{i-1})$. This coordinate is useful for hitting probabilities and <optimal stopping>.