Modeling stationary, periodic, and long memory processes by superposed jump-driven processes

The long memory process is a stochastic process with power-type autocorrelation. Such processes are found worldwide, and those arising in the environmental sciences often have some periodicity, such as seasonal and/or daily timescales. However, unified descriptions of these phenomena have not yet been thoroughly studied. We present a novel mathematical approach for formulating stationary, periodic, and long memory processes based on the superposition of a continuum of jump-driven processes with distributed reversion speeds. Each jump-driven process is vector-valued and analytically tractable, with closed-form characteristic and autocorrelation functions. This tractability is inherited in the resulting superposition process, which facilitates the analysis and application. Moreover, we also address the issue of model uncertainty, where uncertainty is assumed in the probability distribution of the reversion speeds. A divergence to evaluate model uncertainty is proposed based on a unified deformed logarithm along with the associated Orlicz space. The memory characteristic is then explicitly linked to the Orlicz space, using which we can identify the applicability and limitations of divergence through the parameters of the deformed logarithm. As a byproduct, we also investigate some modern divergences found in machine learning methods. The proposed superposition process and divergence are applied to time-series data of dam operations involving hydropeaking outflow.