We study the asymptotics of the ruin probability for a process which is the solution of a linear SDE defined by a pair of independent Lévy processes. Our main interest is a model describing the evolution of the capital reserve of an insurance company selling annuities and investing in a risky asset. Let \(\beta >0\) be the root of the cumulant-generating function \(H\) of the increment \(V_{1}\) of the log-price process. We show that the ruin probability admits the exact asymptotic \(Cu^{-\beta }\) as the initial capital \(u\to \infty \), assuming only that the law of \(V_{T}\) is non-arithmetic without any further assumptions on the price process.

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Lévy驱动的广义Ornstein-Uhlenbeck过程的破产概率

我们研究一个过程的破产概率的渐近性，该过程是由一对独立的Lévy过程定义的线性SDE的解。我们的主要兴趣是一个模型，该模型描述了出售年金并投资风险资产的保险公司的资本准备金的演变。令\（\ beta> 0 \）为对数价格过程的增量\（V_ {1} \）的累积量生成函数\（H \）的根。我们表明，破产概率承认精确渐近\（CU ^ { - \测试} \）作为初始资本\（U \到\ infty \），假设只有法\（V_横置\）是非算术，无需对价格过程进行任何进一步假设。