  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, cf. [22, Corollary 2.9]. 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.
