A series expansion formula of the scale matrix with applications in CUSUM analysis

We introduce a new Lévy fluctuation theoretic method to analyze the cumulative sum (CUSUM) procedure in sequential change-point detection. When observations are phase-type distributed and the post-change distribution is given by exponential tilting of its pre-change distribution, the first passage analysis of the CUSUM statistic is reduced to that of a certain Markov additive process. We develop a novel series expansion formula of the scale matrix for Markov additive processes of finite activity, and apply it to derive exact expressions of the average run length, average detection delay, and false alarm probability under the CUSUM procedure.