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 $R \to \infty$.