My question is related to the paper "Sequential Testing for Poisson Processes" by Peskir and Shiryaev, available here.
Specifically, suppose that there are two states of the world, G(ood) and B(ad), and the agent wants to take a correct action but doesn't know the state of the world. She does have some prior that the state is G, say, p.
Starting at t=0, she begins observing "news". In state G, news are modeled by a Poisson process with a known parameter $\lambda$. In state B, no news is ever observed (say, it's a Poisson process with parameter 0).
Perskir and Shiryav solve the problem for sequential testing between two Poisson processes with parameters $\lambda_1 > \lambda_0 > 0$. As far as I can tell, their method isn't directly applicable for when $\lambda_0 = 0$.
I have an intuition, which is somewhat confirmed by trying to solve the discrete-time case, that the agent's strategy would be characterized by a cutoff time for not observing any news. Obviously, once an arrival is observed, her belief would drift to 1, and the correct action for state G will be taken. If no signal is observed for long enough, she would take the action for state B. I am having trouble formalizing this solution and finding the value of that cutoff. Any help would be greatly appreciated!
