An Optimal Selection Problem Associated with the Poisson Process

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 kth trial is a success with probability pₖ=/ (+k-1) and parameter > 0. In the best-choice setting (=1), the myopic strategy is optimal, and the proof hinges on verifying the monotonicity of certain critical roots. We establish this crucial result in the last-success case by exploiting a connection to the sign of the derivative in the first parameter of a quotient of Kummer's hypergeometric functions. Furthermore, we address the case of inhomogeneous arrivals by employing a point process of success events. We illustrate this approach using an example that combines an inhomogeneous intensity function with a continuous-time success profile. Finally, we derive bounds and asymptotics of the critical roots, strengthening and improving the findings of Ciesielski and Zabczyk (1979).