On optimal dividends with penalty payments in the Cramér-Lundberg model (Q1689031)

Consider a classical Cramér-Lundberg risk model \(X_t^0\). A dividend is paid and negative surplus is allowed. Then the surplus becomes \(X_t^D = X_t^0 - D_t\), where the accumulated dividends \(D\) is a non-decreasing process such that \(D_{0-} = 0\). Higher surplus is preferred to lower surplus, thus a penalty \(\phi(X_t^D)\) is added, and the value of the dividend strategy becomes \[ V^D(x) = E\Bigl[ \int_0^\infty e^{-\delta t} \;d D_t - \int_0^\infty e^{-\delta t} \phi(X_t^D)\;d t\Bigr]\;. \] \(\delta> 0\) is a preference parameter giving more weight to the present than to the far future. The goal is to maximize \(V^D\); \(V(x) = \sup_D V^D(x)\). The penalty function should be continuous, non-increasing, positive and convex. Under these conditions, the optimal dividend strategy is a barrier strategy. \(V(x)\) fulfills the Hamilton-Jacobi-Bellman equation \[ \max \Bigl\{ c V'(x) + \lambda \int_0^\infty V(x-y)\;d F(y) - (\lambda+\delta) V(x) - \phi(x),\; 1 - V'(x)\Bigr\} = 0\;, \] where \(c\) is the premium rate, \(\lambda\) is the claim intensity and \(F\) is the claim size distribution. As examples, exponentially distributed claim sizes in the cases \(\phi(x) = \alpha e^{-\beta x}\) and \(\phi(x) = -\alpha x 1_{x < 0}\) are explicitly calculated.