CUSUM-statistics and its optimality in Lorden's criterion

If $\mathsf{P}^{\theta}$, $\theta\in[0,\infty]$ is a family of probability measures, which are locally absolutely continuous in the measure $\mathsf{P}^{\infty}$, then the quantity $\frac{d\mathsf{P}^{\theta}}{d\mathsf{P}^{\infty}}$, the likelihood ratio, is well known. The value of
$$\gamma_t = \sup_{\theta\leqslant t}\frac{d\mathsf{P}^{\theta}}{d\mathsf{P}^{\infty}}(0,t)\ \ $$
is called the CUSUM-statistics (CUSUM = cumulative sum). For the disorder problem Lorden proposed the following criterion of optimality
$$D = \inf_{\tau\geqslant0}\sup_{\theta\geqslant0} \mathop{\mathrm{ess\,sup}}_{\omega}\mathrm{E}^{\theta}\left((\tau-\theta)^{+}|\mathcal{F}_{\theta}\right)(\omega),$$
where $\tau$ is stopping time.
In the talk it will be discussed how, for this criterion, (in the case of the Brownian motion whose drift is changed at the moment $\theta$) the CUSUM-optimality is proved.