I am looking for a reference that covers the following special case of quickest detection (or sequential hypothesis testing) problem. I believe there should be a large literature on it either in mathematical finance or statistical decision theory.
Let $\theta$ be unobserved. The decision maker (DM) would like to choose action $A_1$ if $\theta>0$ and $A_2$ otherwise. Before making decision, the DM can choose to wait to learn more about $\theta$, but this is costly. The DM is choosing when to stop learning and choose an action.
Thus, this is an optimal stopping time of the form \begin{align} \inf_{\tau} \mathbb{E}\left[g(\pi_{\tau}) - c\tau|\pi_0\right], \end{align} where $\pi_t$ is the DM's belief at time $t$ based on information DM received ($\pi_0$ is the prior belief) and $g(\pi)$ is the expected payoff of making a decision when the belief is $\pi$.
I am looking for books/notes (or accessible papers) that cover these types of problems. I am particularly, interested in the case where $\theta~\sim N(\mu,\sigma^2)$ and DM is observing process $dY_t=\theta dt + \nu dW_t,$ with $W_t$ a standard brownian motion.
I think the most important reference on the Quickest Detection is the book is "Quickest Detection" by H. Vincent Poor, Cambridge University Press. Here you can find a list of the subject covered by the book https://www.cambridge.org/core/books/quickest-detection/4A7686F64138F460CC10148FC58E52F8
Regards,
Andrea