If there wouldn’t be a delay between observing the current value and stopping (assuming a reasonably well-behaved distribution P), that problem is trivial: stop

If there wouldn’t be a delay between observing the current value and stopping (assuming a reasonably well-behaved distribution P), that problem is trivial: stop when you reach the threshold. So, you would have to add some time delay between observation of a value and actual stopping. Intuitively, I can think of two stopping strategies that might be optimal: call ‘stop’ when the expected value at stop is that ≈ 0.5706 or call stop when the probability of overshooting it is 50%. I’m fairly sure that intuition is wrong in general, though.