论文标题
关于通过Lindley型递归的波动理论分解
On fluctuation-theoretic decompositions via Lindley-type recursions
论文作者
论文摘要
考虑在指数分布的时间$t_β$，平均$ 1/β$上的Lévy流程$ y（t）$。我们研究运行最大$ \ bar {y}（t_β）$的联合分布以及该最大值最后一次发生的时间时期$ g（t_β$）。 Our main result is a fluctuation-theoretic distributional equality: the vector ($\bar{Y}(T_β),G(T_β)$) can be written as a sum of two independent vectors, the first one being ($\bar{Y}(T_{β+ω}),G(T_{β+ω})$) and the second one being the running maximum and corresponding time epoch under the restriction仅在Poisson（$ω$）检查时期（直到$t_β$）才观察到Lévy过程。我们首先为这种显着的分解提供了一个分析证明，然后提供了一个更基本的证据，该证明可以深入了解分解的发生，以及$ω$仅出现在分解的右侧。基本推导的基础证明技术也导致了分解的进一步概括，以及对众所周知的林德利递归的概括的一些基本见解。
Consider a Lévy process $Y(t)$ over an exponentially distributed time $T_β$ with mean $1/β$. We study the joint distribution of the running maximum $\bar{Y}(T_β)$ and the time epoch $G(T_β$) at which this maximum last occurs. Our main result is a fluctuation-theoretic distributional equality: the vector ($\bar{Y}(T_β),G(T_β)$) can be written as a sum of two independent vectors, the first one being ($\bar{Y}(T_{β+ω}),G(T_{β+ω})$) and the second one being the running maximum and corresponding time epoch under the restriction that the Lévy process is only observed at Poisson($ω$) inspection epochs (until $T_β$). We first provide an analytic proof for this remarkable decomposition, and then a more elementary proof that gives insight into the occurrence of the decomposition and into the fact that $ω$ only appears in the right hand side of the decomposition. The proof technique underlying the more elementary derivation also leads to further generalizations of the decomposition, and to some fundamental insights into a generalization of the well known Lindley recursion.
