EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE INVERSE BOUNDARY CROSSING PROBLEM FOR DIFFUSIONS
EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE INVERSE BOUNDARY CROSSING PROBLEM FOR DIFFUSIONS
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DOI：
10.1214/10-aap714
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发表时间：
2011-10-01
影响因子：
1.8
通讯作者：
Saunders, David
中科院分区：
文献类型：
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作者：
Chen, Xinfu;Cheng, Lan;Saunders, David
我们研究了扩散的反边界穿越问题。在给定扩散过程X-t和[0，8)上的生存分布p的情况下，我们证明了存在一个边界b(T)使得p(T)=P[tau>；t]，其中t是X-t到达边界b(T)的第一次命中时间。所采用的方法是解析的，基于求解一个抛物型变分不等式来寻找b。在以前的工作中证明了该变分不等式解的存在唯一性。在这篇文章中，我们证明了所得到的边界b确实有p作为它的边界交叉分布。由于对由变分不等式引起的b的正则性知之甚少，这就需要详细研究计算X-t到粗边界的边界穿越分布的问题。给出了用相应的Kolmogorov正演方程的弱解表示这一问题的结果。
We study the inverse boundary crossing problem for diffusions. Given a diffusion process X-t, and a survival distribution p on [0,8), we demonstrate that there exists a boundary b(t) such that p(t) = P [tau > t], where t is the first hitting time of X-t to the boundary b(t). The approach taken is analytic, based on solving a parabolic variational inequality to find b. Existence and uniqueness of the solution to this variational inequality were proven in earlier work. In this paper, we demonstrate that the resulting boundary b does indeed have p as its boundary crossing distribution. Since little is known regarding the regularity of b arising from the variational inequality, this requires a detailed study of the problem of computing the boundary crossing distribution of X-t to a rough boundary. Results regarding the formulation of this problem in terms of weak solutions to the corresponding Kolmogorov forward equation are presented.