In this talk we take a general look at problems of optimal stopping on a success in (nonhomogeneous) Bernoulli trials, in the context of their coupling with stopping of a Poisson process with only time-dependent payoff.

For many known examples the embedding in continuous time adds considerable insight, allowing one to replace cumbersome combinatorial calculations of the optimum by simple approximate solutions. For both Bernoulli and Poisson frameworks, we characterise the optimality of single-threshold rules by a simple unimodality condition, and for the general reward functions suggest an efficient method to tackle the problems with stopping islands.