The Last-Success Stopping Problem with Random Observation Times

Suppose $N$ independent Bernoulli trials are observed sequentially at random times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. We focus on the version of the problem where the $k^\text{th}$ trial is a success with probability $p_k=\theta/(\theta+k-1)$ and the prior distribution of $N$ is negative binomial with shape parameter $\nu$. Exploring properties of the Gaussian hypergeometric function, we find that the myopic stopping strategy is optimal if and only if $\nu\geq\theta$. We derive formulas to assess the winning probability and discuss limit forms of the problem for large $N$.