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Principle Seminar of the Department of Probability Theory, Moscow State University
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CUSUM-statistics and its optimality in Lorden's criterion A. N. Shiryaevab a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow b Lomonosov Moscow State University, Faculty of Mechanics and Mathematics |
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Abstract: If $$\gamma_t = \sup_{\theta\leqslant t}\frac{d\mathsf{P}^{\theta}}{d\mathsf{P}^{\infty}}(0,t)\ \ $$ 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 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 |