Investigating Mean-Reverting Processes with Gumbel Noise
Fatimah Alshahrani, Department of Statistics and Probability, Michigan State University, USA
Extreme Value Theory (EVT) deals with studying "rare events" by taking either the maxima or minima of observations. We applied EVT to the annual maximum for sea level data provided by the Actuaries Climate Index, (ACI). Our analysis shows that the annual maxima follow the Type I Generalized Extreme Value Distribution, which is known as the Gumbel distribution. The time series of monthly records, as described by ACI is said to follow the Gumbel distribution. Initial investigation shows the de-trended data has some of the features of the Ornstein Uhlenbeck (OU) process, i.e. a Gaussian mean reverting process. However, unlike the OU process, our data is presumably non- Gaussian. Using the relationship between stochastic differential equations and discrete time processes, we are building a reliable model using tools from stochastic analysis including the Malliavin Calculus, and objects such as AR(1) process and the OU process, rather than relying upon the so-called Gumbel noise, which is an object used largely in Machine Learning.