Controlling the rain fall statistics using Mean-Reverting Jump Diffusion model

Abstract

We present a stochastic mean-reverting jump-diffusion model to simulate rainfall time series and validate it using long-term half-hourly rain fall data from the North-East region of India. The model captures the intermittent and extreme-event dynamics of rainfall, reproducing superdiffusive behavior with an exponent $\sim 1.8$, along with the observed probability distributions and multifractal features. By systematically varying key parameters, we demonstrate a transition between Log-Normal and Gamma distributions, and show how the occurrence of extreme events and dry-patch durations can be controlled. Spectral and wavelet analyses further confirm that the simulated series reproduces the dominant temporal scales observed in real rainfall data. Our proposed framework provides a robust tool for generating realistic synthetic rainfall series and serves as an effective approach for understanding the influence of underlying stochastic processes that governs the rainfall statistics.