Period-two martingale corrector for a birth-death chain ($h_{\theta }(k)=k-\theta 1_{\{k\text{ odd}\}}$)

For a birth-death chain on the integers with right/left probabilities $p_i,q_i>0$ depending on parity $i\in \{0,1\}$, define $h_{\theta }(k)=k-\theta 1_{\{k\text{ odd}\}}$. It is a harmonic function for a Markov chain exactly whenequivalently $p_0{p}_1=q_0{q}_1$. The correction is bounded. Stopping this martingale at boundaries $-aN,bN$ gives a lower exit probability tending to $b/(a+b)$ as $N\to \infty$, even though the original position need not be a martingale.

// Target: markov-process.bigb