Asymptotic exponentiality of the first exit time of the Shiryaev–Roberts diffusion with constant positive drift. Issue 3 (3rd July 2017) - Polunchenko, Aleksey S.
ABSTRACT: We consider the first exit time of a Shiryaev–Roberts diffusion with constant positive drift from the interval [0, A ] with A >0 fixed. We show that the moment generating function (Laplace transform) of a suitably standardized version of the first exit time converges to that of the unit-mean exponential distribution as A →+∞. The proof is explicit in that the moment generating function of the first exit time is first expressed analytically and in a closed form and then the desired limit as A →+∞ is evaluated directly. The result is of importance in the area of quickest change-point detection, and its discrete-time counterpart has been previously established—although in a different manner—by Pollak and Tartakovsky (2009a ).