  We consider the first exit time of a Shiryaev-Roberts diffusion with constant
positive drift from the interval $[0,A]$ where $A>0$. 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 $A\to+\infty$. 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 $A\to+\infty$ 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 (2009).
