In this paper we consider a natural generalization in continuous time of the so-called
secretary problem. This well-known stopping-time problem is discussed by Gardner [1] and
Dynkin [2] and has a large literature dating back to Cayley [3] in 1875. The problem that we
shall consider is as follows. A man has been allowed a fixed time T in which to find an
apartment. Opportunities to inspect apartments occur at the epochs of a stationary Poisson
process of intensity A. He inspects each apartment immediately the opportunity arises; …

Cowan and Zabczyk (1978) introduced a continuous-time generalisation of the secretary
problem, where offers arrive at epochs of a homogeneous Poisson process. We expand
their work to encompass the last success problem under the Karamata-Stirling record profile.
In this setting, the $ k $ th trial is a success with probability $ p_k=\theta/(\theta+ k-1) $ and
parameter $\theta> 0$. In the best choice setting ($\theta= 1$), the myopic strategy is
optimal, and the proof hinges on verifying the monotonicity of certain critical roots. We …