Change detection

Problem

Let $X_1, X_2, \dots$ be observed one at a time. For an unknown change time $\nu \in \set{1, 2, \dots} \cup \set{\infty}$, the variables are independent, $X_n$ has density $f_0$ for $n < \nu$ and density $f_1$ for $n \ge \nu$, where $f_0 \ne f_1$. Find a stopping time $T$ of the observations that raises an alarm soon after the change and rarely before it.

Lorden's minimax form measures the delay by $$\bar D(T) = \sup_{\nu \ge 1} \operatorname*{ess\,sup} E_\nu\left[ (T - \nu + 1)^+ \mid X_1, \dots, X_{\nu - 1} \right],$$ and the task is to minimize $\bar D(T)$ subject to $E_\infty[T] \ge \gamma$ for a given $\gamma > 1$. Here $E_\nu$ is the expectation when the change occurs at time $\nu$, and $E_\infty$ is the expectation when there is no change.