State-dependence of noise (multiplicative noise) is crucial: when PP is already high or TT low, identical shocks have bigger social impact---captured by letting GG grow with PP and with (1T)(1-T).
2. Noise-induced tipping and mean first-passage times
In deterministic bifurcation theory, transitions occur as parameters cross critical values. With noise, two additional phenomena arise:
Noise-induced tipping: even if control parameters \theta are subcritical (<c\mu<\mu_c), sufficiently large or persistent noise can push the system across basins of attraction.
Stochastic resonance and coherence resonance: moderate noise can amplify oscillatory modes (if near a Hopf bifurcation), producing intermittent waves of unrest.
A practical metric: mean first-passage time (MFPT) to a critical threshold (e.g., PP crossing PcP_c or HH crossing HcH_c). MFPT can be estimated numerically by Monte Carlo simulation of the SDE ensemble or approximated analytically in low-noise regimes via Kramers' escape formula (generalized to multi-dimensional systems via large deviation theory).
3. Parameter sensitivity: local and global methods
Parameter uncertainty is inevitable. To produce operational forecasts we must quantify how model outputs (probabilities of crossing thresholds in given windows, steady-state means, autocorrelation indicators) depend on parameters.
Approaches:
Local sensitivity (linearized): compute Jacobian sensitivities x/\partial x^\ast/\partial \theta and finite differences of output metrics for small perturbations. Useful for quick diagnostics and to identify the most influential parameters near an operating point.
Global sensitivity (variance-based): e.g., Sobol indices or Fourier Amplitude Sensitivity Testing (FAST). These decompose output variance over the full parameter ranges and identify interactions and nonlinear sensitivities. Particularly important when parameters (e.g., K,U,P,E,\rho_{\mathcal{K}},\rho_U,\alpha_P,\theta_E,\kappa) interact to produce bifurcation behavior.
Screening methods: Morris method to rank parameters when computational budgets are limited.
Recommended parameter sets to probe thoroughly:
Policy responsiveness: U,K\rho_U,\rho_{\mathcal{K}} (and ratio rr).
Trust and decay rates: U,K,T\eta_U,\eta_{\mathcal{K}},\lambda_T.
Economic coupling: P,U,E,E,\phi_P,\phi_U,\mu_E,\theta_E,\kappa.
Black-horse growth: P,E,T,H\alpha_P,\alpha_E,\beta_T,\gamma_H.
Noise amplitudes: entries of G()G(\cdot) and jump rate/size.
4. Practical calibration and uncertainty quantification
