A note on optimal stopping for possible change in the intensity of an ordinary Poisson process

Peskir and Shiryaev [2002. Solving the Poisson disorder problem. In: Advances in Finance and Stochastics: Essays in Honor of Dieter Sonderman. Springer, New York, pp. 295-312] determined the optimal stopping rule for a problem of quick detection of a change-point in the intensity of a homogeneous ordinary Poisson process, when the cost per unit time of delayed detection is in a given range, and the change-point occurs at random times following a mixed exponential distribution. Using the same Bayesian framework, we extend their results to a range of cost values not considered before. We obtain the results by using the Dynamic Programming rather than the analytical methods used by Peskir and Shiryaev.