Learning to Detect Events with Markov-Modulated Poisson Processes

[tangbotony]

Excuse me, After reading Learning to Detect Events with Markov-Modulated，I have some doubts. In the 6.2 section, I think the formula

should be

Because we need to think about Z(t)=-1. I’m not sure whether I understand it correctly. Can you explain this formula? Thank you.