Stochastic Differential Equation of Earthquakes Series

Abstract

This work is devoted to modeling earthquake time series. We propose a stochastic differential equation based on the superposition of independent Ornstein–Uhlenbeck processes driven by a $$\Gamma (\alpha, \beta )$$ process. Superposition of independent $$\Gamma (\alpha, \beta )$$ Ornstein–Uhlenbeck processes offer analytic flexibility and provides a class of continuous time processes capable of exhibiting long memory behavior. The stochastic differential equation is applied to the study of earthquakes by fitting the superposed $$\Gamma (\alpha, \beta )$$ Ornstein–Uhlenbeck model to earthquake sequences in South America containing very large events (Mw $$\ge $$ 8). We obtained very good fit of the observed magnitudes of the earthquakes with the stochastic differential equations, which supports the use of this methodology for the study of earthquakes sequence.