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We consider the first exit time of a Shiryaev-Roberts diffusion with constantpositive drift from the interval $[0,A]$ where $A>0$. We show that the momentgenerating function (Laplace transform) of a suitably standardized version ofthe first exit time converges to that of the unit-mean exponential distributionas $A\to+\infty$. The proof is explicit in that the moment generating functionof 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 resultis of importance in the area of quickest change-point detection, and itsdiscrete-time counterpart has been previously established - although in adifferent manner - by Pollak and Tartakovsky (2009).

Asymptotic Exponentiality of the First Exit Time of the Shiryaev-Roberts Diffusion with Constant Positive Drift