Synthesis of a Short‐Memory Generalized Extreme Value–Based Stochastic Differential Equation With Application to Rainfall Extremes

Extreme value theory (EVT) is the standard framework for modeling rare events, but most applications remain static and do not explicitly represent temporal dependence. This study develops a synthesis‐based stochastic differential equation (SDE) whose stationary distribution is based on the generalized extreme value (GEV) law. The main methodological contribution is the extension of synthesis‐based diffusion modeling to the full GEV family, including the Gumbel limit, through explicit derivation of drift and diffusion functions from the GEV density. The resulting diffusion process preserves the marginal extreme‐value structure while embedding mean‐reverting short‐memory dynamics through an interpretable correlation‐time parameter. The framework is validated using annual maximum rainfall from four regions of Tanzania: Rukwa, Iringa, Mbeya, and Ruvuma. Region‐specific GEV parameters are estimated by maximum likelihood and used to construct corresponding SDEs. Model performance is assessed through histogram comparisons, log–log tail diagnostics, Q–Q envelopes, autocorrelation analysis, return‐level evaluation, and sequential in‐sample/out‐of‐sample forecasting. Results show that the synthesized SDE reproduces the fitted marginal distribution and captures the weak persistence structure of the observed extremes. Predictive interval coverage remains close to nominal levels, and forecast error measures indicate stable performance across climatically heterogeneous regions. These findings show that the proposed GEV‐based SDE provides a practically interpretable bridge between static EVT inference and dynamic stochastic modeling for hydrological risk assessment. More broadly, the synthesis framework offers a useful basis for simulation, forecasting, and uncertainty quantification, while simulations targeting genuine long‐range dependence remain a topic for future research.