An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries.
An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries.
复制标题
半鞅的不变性原理反映了具有分段平滑边界的域中的布朗运动。
DOI：
10.1214/105051606000000899
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发表时间：
2007
影响因子：
1.8
通讯作者：
R. J. Williams
中科院分区：
文献类型：
--
作者：
W. Kang;R. J. Williams
反映布朗运动 (SRBM) 的布朗运动 (SRBM) 生活在具有分段平滑边界的域的闭包中，它在应用概率中引起了人们的兴趣，因为它们在某些随机网络中作为大流量近似的作用。在本文中，假设反射域和方向上的某些条件，证明了短程弹道导弹的摄动结果或不变性原理。这为满足 SRBM 定义的过程提供了充分的条件，除了定义条件中的小随机扰动之外，其分布接近 SRBM。证明这一结果的一个关键因素是扰动 Skorokhod 问题解的振荡不等式。我们利用不变性原理证明了短程弹道导弹在温和条件下的弱存在性。我们还使用不变性原理，结合已知的 SRBM 唯一性结果，给出一些充分条件来验证近似值，其中涉及（i）凸多面体中的 SRBM，在多面体的每个面上具有恒定的反射矢量场，以及（ii）有界域中的 SRBM，其具有分段平滑边界以及边界面上可能存在非恒定反射矢量场。
Semimartingale reflecting Brownian motions (SRBMs) living in the closures of domains with piecewise smooth boundaries are of interest in applied probability because of their role as heavy traffic approximations for some stochastic networks. In this paper, assuming certain conditions on the domains and directions of reflection, a perturbation result, or invariance principle, for SRBMs is proved. This provides sufficient conditions for a process that satisfies the definition of an SRBM, except for small random perturbations in the defining conditions, to be close in distribution to an SRBM. A crucial ingredient in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. We use the invariance principle to show weak existence of SRBMs under mild conditions. We also use the invariance principle, in conjunction with known uniqueness results for SRBMs, to give some sufficient conditions for validating approximations involving (i) SRBMs in convex polyhedrons with a constant reflection vector field on each face of the polyhedron, and (ii) SRBMs in bounded domains with piecewise smooth boundaries and possibly nonconstant reflection vector fields on the boundary surfaces.