非负正交点上Skorohod反射映射初始条件的渐近无关性
Asymptotic irrelevance of initial conditions for Skorohod reflection mapping on the nonnegative orthant
摘要：
将由非负正交体上的确定性Skorohod问题所引起的反射映射应用于$[0,\infty)$上的$\mathbb{R}^n$值函数$X$，然后应用于$a+X$，其中$a$是非负常数向量。一个已经开放了15年的问题是，对于任何$a$的选择，随着时间的推移，在什么条件下两个所得调节函数之间的差收敛到零。这又意味着，如果施加足够的随机结构，确保应用于多维过程$X$的反射映射在分布上收敛，那么当它应用于$η+X$时，它也将在分布上收敛，其中$η$是甚至可能依赖于过程$X$的任何几乎肯定有限值的随机向量。在本文中，我们得到了一个有用的等价刻画这个属性。因此，我们能够确定一个自然的充分条件，在给定的数据$X$和常数路由矩阵。还指出了类似的必要条件。我们的分析的一个特殊含义是，在额外的随机假设下，初始条件的渐近无关性不需要存在一个平稳分布。作为我们的（和以前的）结果的直接推论，我们得出结论，在自然稳定条件下，反射Lévy过程以及马尔可夫加性过程具有唯一的平稳分布，并且对于每个初始条件，在分布上收敛于该平稳分布。扩展的充分条件，然后开发的反射映射与漂移和路由系数，可能是时间和状态依赖的一些影响，多维保险模型进行了简要讨论。
英文摘要：
A reflection map, induced by the deterministic Skorohod problem on the nonnegative orthant, is applied to an $\mathbb{R}^n$ valued function $X$ on $[0,\infty)$ and then to $a+X$, where $a$ is a nonnegative constant vector. A question that has been open for over 15 years is under what conditions the difference between the two resulting regulated functions converges to zero for any choice of $a$ as time diverges. This in turn implies that if one imposes enough stochastic structure that ensures that the reflection map applied to a multidimensional process $X$ converges in distribution then it will also converge in distribution when it is applied to $η+X$ where $η$ is any almost surely finite valued random vector that may even depend on the process $X$. In this paper we obtain a useful equivalent characterization of this property. As a result we are able to identify a natural sufficient condition in terms of the given data $X$ and the constant routing matrix. A similar necessary condition is also indicated. A particular implication of our analysis is that under additional stochastic assumptions, asymptotic irrelevance of the initial condition does note require the existence of a stationary distribution. As immediate corollaries of our (and earlier) results we conclude that under the natural stability conditions, a reflected Lévy process as well as Markov additive process has a unique stationary distribution and converges in distribution to this stationary distribution for every initial condition. Extensions of the sufficient condition are then developed for reflection maps with drift and routing coefficients that may be time and state dependent; some implications to multidimensional insurance models are briefly discussed.
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