Record Breaking Events in Stochastic Processes

A record is an entry in a time series that is larger (upper record) or smaller (lower record) than all previous entries. In financial mathematics record events play a big role for example when we think of changes in stock prices. Obviously not only the probability that a record occurs is of interest but also the correlations between record events. How likely is that one record is followed by another one immediately? The financial crisis is a good example to be considered, since consecutive lower records may occur. Realizing such an extreme event in its early stages can prevent heavy loss by reacting appropriately. What you should do is drop the affected financial products. In this work we consider four stochastic processes: the Wiener process with drift, the geometric Brownian motion, the Ornstein-Uhlenbeck process and the Cox-Ingersoll-Ross model. We calculate the probability Pn that all n entries in a time series are records, the probability pn that the nth entry is a record and the joint probability pn,n−1 that the (n − 1)th entry is a record as well as the nth entry, enabling us to consider correlations between record events. Finally we divide pn,n−1 by the probabilities pn and pn−1 to get the normalized joint probability ln,n−1. We perform a distributionfree method based on the record correlations to detect heavy-tail behavior. We show that the method works well for small dataset. The maximum likelihood method for estimating the parameters of the processes and the model validation and selection are part of this work, too