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 \(R \to \infty\).
We study the optimal stopping problem in continuous time, where an agent chooses a stopping time to maximise the expected value of a payoff process, following the classical framework of Snell (1952) and its continuous-time extension via the Doob–Meyer decomposition. The value function is characterised as the Snell envelope — the smallest supermartingale dominating the payoff — whose generator satisfies a Hamilton– Jacobi–Bellman variational inequality of obstacle type. The optimal stopping time is the first entry into the stopping region, where the value function equals the payoff, and the continuation region is determined by the strict inequality V > g. As a canonical application, we solve the perpetual American put option, obtaining the closed-form exercise boundary and value function, and illustrate how the exercise boundary moves with time-to-expiry under the finite-horizon formulation.
The Russian option, introduced by Shepp and Shiryaev (1993), is a perpetual American lookback contract whose payoff is the running maximum of the underlying asset. We price it via two independent frameworks: a Hamilton-Jacobi-Bellman (HJB) variational inequality and a singly reflected backward stochastic differential equation (BSDE) with barrier $M_t$. The HJB approach yields a closed-form solution through a scalar Euler ODE, while the BSDE Bellman iteration — solved by Gauss-Hermite quadrature — converges to the same value function with mean relative error $0.10\%$. A third result ties the two together: the expected stopping time $u(x) = \mathbb{E}[\tau^* \mid X_0 = x]$ satisfies a companion Poisson equation $\mathcal{L}u = -1$ under the same operator and boundary conditions, and its closed form reveals that the option premium equals, to leading order, the discount rate times the expected wait times the current value.