AI 中文总结
受网络攻击自激模型启发引入一类具重置的分段确定性马尔可夫过程，证明其不变分布的存在唯一性并推导密度，制定长期平均控制问题，对指数分布跳跃大小的情况可明确表征不变分布及最优干预策略。
AI 中文摘要
我们引入了一类具有重置的分段确定性马尔可夫过程，其灵感来自网络攻击的自激模型。在类似于破产理论的假设下，我们证明了不变分布的存在性和唯一性，并明确推导了其密度，从而确立了该过程的遍历性。然后，我们制定了一个相关的长期平均控制问题，其中重置作为干预机制。对于指数分布的跳跃大小，该模型在分析上易于处理，允许对不变分布和最优干预策略进行明确表征。
英文摘要
We introduce a class of piecewise deterministic Markov processes with resetting, motivated by self-exciting models of cyber attacks. Under assumptions reminiscent of ruin theory, we prove the existence and uniqueness of an invariant distribution and derive its density explicitly, thereby establishing ergodicity of the process. We then formulate an associated long-run average control problem in which resetting acts as the intervention mechanism. For exponentially distributed jump sizes, the model becomes analytically tractable, allowing an explicit characterization of the invariant distribution and of the optimal intervention policy.
