AI 中文总结
本文针对含环境风险的随机增长模型，扩展泊松点过程框架至标记霍克斯过程驱动的灾害，建立受控霍克斯动力学适定性，获霍克斯激发过程定量估计，证明小激发下霍克斯价值函数收敛于泊松对应值，为随机控制问题提供泊松近似并量化自激发影响。
AI 中文摘要
本文致力于构建考虑环境风险的随机增长模型框架，其中罕见但灾难性的冲击与资本积累和污染相互作用。基于arXiv:2511.13568研究的泊松点过程公式，我们将模型扩展到由标记霍克斯过程驱动的灾害，允许过去的灾害暂时增加后续冲击的可能性。本研究聚焦于亚临界马尔可夫霍克斯规范，其状态空间因灾害风险的自激发分量而扩充。我们建立了所得受控霍克斯动力学的适定性和非爆炸特性。利用对应泊松控制问题的哈密顿-雅可比-贝尔曼表征，我们获得了霍克斯激发过程的定量估计，并证明在小激发尺度下，当自激发幅度消失时，霍克斯价值函数收敛到其泊松对应值。这为随机控制问题提供了严格的泊松近似，并量化了环境灾害风险下自激发对最优增长的影响。
英文摘要
This paper is devoted to developing a framework for stochastic growth models with environmental risk, in which rare but catastrophic shocks interact with capital accumulation and pollution. Building on the Poisson point process formulation studied in arXiv:2511.13568, we extend the model to disasters driven by a marked Hawkes process, allowing past disasters to temporarily increase the likelihood of subsequent shocks. Our work focuses on a subcritical Markovian Hawkes specification, in which the state space is augmented by the self-excitation component of disaster risk. We establish the well-posedness and nonexplosion of the resulting controlled Hawkes dynamics. Using the Hamilton-Jacobi-Bellman characterization of the corresponding Poisson control problem, we obtain quantitative estimates for the Hawkes excitation process and prove, under a small-excitation scaling, that the Hawkes value function converges to its Poisson counterpart as the magnitude of self-excitation vanishes. This provides a rigorous Poisson approximation of the stochastic control problem and quantifies the effect of self-excitation on optimal growth under environmental disaster risk.
