Exact Optimal Stopping for Multidimensional Linear Switching Diffusions
研究了一类多维线性切换扩散过程在无限时间区间上的最优停时问题，允许成本函数不可积，证明了停时区域的首达时间是最优的，并将最优边界表示为非线性积分方程的唯一解，最后应用于马尔可夫漂移的快速实时检测。
The paper studies a class of multidimensional optimal stopping problems with infinite horizon for linear switching diffusions. There are two main novelties in the optimal problems considered: The underlying stochastic process has discontinuous paths, and the cost function is not necessarily integrable on the entire time horizon, where the latter is often a key assumption in classical optimal stopping theory for diffusions. Under relatively mild conditions, we show, for the class of multidimensional optimal stopping problems under consideration, that the first entry time of the stopping region is an optimal stopping time. Further, we prove that the corresponding optimal stopping boundaries can be represented as the unique solution to a nonlinear integral equation. We conclude with an application of our results to the problem of quickest real-time detection of a Markovian drift. Funding: P. Ernst received financial support from the Army Research Office [Grant ARO-YIP-71636-MA], the National Science Foundation [Grant DMS-1811936], and the Office of Naval Research [Grants N00014-18-1-2192 and N00014-21-1-2672].