The level crossing and inverse statistic analysis of German stock market index (DAX) and daily oil price time series

The level crossing and inverse statistics analysis of DAX and oil price time series are given. We determine the average frequency of positive-slope crossings, $\nu_{\alpha}^+$, where $T_{\alpha} =1/\nu_{\alpha}^+ $ is the average waiting time for observing the level $\alpha$ again. We estimate the probability $P(K, \alpha)$, which provides us the probability of observing $K$ times of the level $\alpha$ with positive slope, in time scale $T_{\alpha}$. For analyzed time series we found that maximum $K$ is about 6. We show that by using the level crossing analysis one can estimate how the DAX and oil time series will develop. We carry out same analysis for the increments of DAX and oil price log-returns,(which is known as inverse statistics) and provide the distribution of waiting times to observe some level for the increments.