AI 中文总结
该研究针对参数依赖马尔可夫随机环境的风险资产投资的保险公司破产问题，用格林函数方法证明年金支付生存概率的C²光滑性，明确了双侧跳跃下光滑性的尖锐条件及二阶导数不连续的计算方式。
AI 中文摘要
我们考虑保险公司将全部准备金投资于参数依赖马尔可夫随机环境的风险资产的破产问题。利用格林函数方法，在最小假设下证明年金支付的生存概率具有C²光滑性——跳跃分布仅需为正半轴上的概率测度。对于双侧跳跃，当每个状态下跳跃分布在负半轴上无原子，或生存概率在原点处消失时，光滑性成立。该条件是尖锐的：孤立原子结合原点处的正值会使二阶导数不连续，且不连续的大小可明确计算。
英文摘要
We consider the ruin problem for an insurance company investing its whole reserve in a risky asset whose parameters depend on a Markov random environment. Using the Green's function method we prove $C^2$-smoothness of the survival probability for annuity payments under minimal assumptions-the jump distribution need only be a probability measure on the positive half-line. For two-sided jumps smoothness holds whenever, in each regime, the jump distribution has no atoms on the negative half-line or the survival probability vanishes at the origin. This condition is sharp: an isolated atom combined with a positive value at the origin makes the second derivative discontinuous, and the size of the discontinuity is computed explicitly.
Comments10 pages
