Spectral Bounds and Exit Times for a Stochastic Model of Corruption

We study a stochastic differential model for the dynamics of institutional corruption, extending a deterministic three-variable system -corruption perception, proportion of sanctioned acts, and policy laxity- by incorporating Gaussian perturbations into key parameters. We prove global existence and uniqueness of solutions in the physically relevant domain, and we analyze the linearization around the asymptotically stable equilibrium of the deterministic system. Explicit mean-square bounds for the linearized process are derived in terms of the spectral properties of a symmetric matrix, providing insight into the temporal validity of the linear approximation. To investigate global behavior, we relate the first exit time from the domain of interest to backward Kolmogorov equations and solve the associated elliptic and parabolic PDEs numerically with FreeFEM, obtaining estimates of expectations and survival probabilities. An application to the case of Mexico highlights nontrivial effects: while the spectral structure governs local stability, institutional volatility can nonmonotonically accelerate global exit, showing that highly reactive interventions without effective sanctions increase uncertainty. Policy implications and possible extensions are discussed.