10-MAY-2025
We formulate the erosion of social accountability (Omega(t)) under growing private wealth (R(t)) as a stochastic optimal control problem on (mathbb{R}_+). The wealth process follows a pure Kou double-exponential jump model with no diffusion and no drift: two independent compound Poisson processes with rates (lambda_1, lambda_2) and exponential jump sizes (mathrm{Exp}(eta_1)), (mathrm{Exp}(eta_2)), capturing large upward and small downward jumps respectively. We derive the Hamilton-Jacobi-Bellman integro-differential equation for the value function (V(R,Omega)), convert it to a Fredholm integral equation of the second kind with bilateral exponential kernel (K(R,x)), and identify the Wiener-Hopf structure on the half-line via the rational symbol (Phi(xi) = 1 – hat{k}(xi)). The Kou model’s quadratic numerator yields an analytic factorisation (Phi = Phi^+ Phi^-), reducing the problem to a second-order ODE whose characteristic roots (beta_1, beta_2) are the Cramér-Lundberg exponents, giving the explicit solution (V(R) = A,e^{-zeta R} + V_p(R)) with (zeta = |beta_2|). The optimal control is bang-bang: a justice curve (mathcal{J} = {R^gamma eta = Omega P}) separates prosocial from antisocial behaviour, and above (mathcal{J}) accountability collapse (Omega to 0) is the rational optimum as (R to infty).