In this paper we consider the optimal dividend problem for an insurance company whose surplus follows a classical Cramér-Lundberg process with a feature of self-exciting. A Hawkes process is applied so that the occurrence of a jump in the claims triggers more sequent jumps. We show that the optimal value function is a unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation with a given boundary condition and declare its concavity. We introduce a barrier curve strategy and verify its optimality. Finally, some numerical results are exhibited.

中文翻译：

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具有传染性的索赔到达的保险公司的最优股息政策

在本文中，我们考虑了一家保险公司的最优股利问题，该公司的盈余遵循经典的Cramér-Lundberg过程，具有自激特性。应用霍克斯（Hawkes）过程，以使索赔中出现跳变会触发更后续的跳变。我们证明了最佳值函数是在给定边界条件下相关联的Hamilton-Jacobi-Bellman方程的唯一粘度解，并声明了其凹度。我们引入了障碍曲线策略并验证了其最优性。最后，给出了一些数值结果。